Surrogate modelling of a detailed farm‐level model using deep learning

Shang, Linmei e al.
A icle — Published Ve sion
Su oga e modelling o a de ailed a m‐le el model using
deep lea ning
Jou nal o Ag icul u al Economics
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Leibniz Ins i u e o Ag icul u al De elopmen in T ansi ion Economies (IAMO), Halle (Saale)
Sugges ed Ci a ion: Shang, Linmei e al. (2024) : Su oga e modelling o a de ailed a m‐le el model
using deep lea ning, Jou nal o Ag icul u al Economics, ISSN 1477-9552, Wiley, Hoboken, NJ, Vol. 75,
Iss. 1, pp. 235-260,
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DOI: 10.1111/1477-9552.12543
ORIGINAL ARTICLE
Su oga e modelling o a de ailed a m- le el model
using deep lea ning
LinmeiShang1 | Ji engWang2 | Da idSchä e 1 |
ThomasHeckelei1 | Jue genGall2,3 | F anziskaAppel4 |
HugoS o m1
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion- NonComme cial- NoDe i s License, which
pe mi s use and dis ibu ion in any medium, p o ided he o iginal wo k is p ope ly ci ed, he use is non- comme cial and no
modi ica ions o adap a ions a e made.
© 2023 The Au ho s. Jou nal o Ag icul u al Economics published by John Wiley & Sons L d on behal o Ag icul u al Economics
Socie y.
1Ins i u e o Food and Resou ce Economics
(ILR), Uni e si y o Bonn, Bonn, Ge many
2Depa men o In o ma ion Sys ems and
A i icial In elligence, Uni e si y o Bonn,
Bonn, Ge many
3Lama Ins i u e o Machine Lea ning
and A i icial In elligence, Sank Augus in,
Ge many
4Leibniz Ins i u e o Ag icul u al
De elopmen in T ansi ion Economies
(IAMO), Halle (Saale), Ge many
Co espondence
Linmei Shang, Ins i u e o Food and
Resou ce Economics (ILR), Uni e si y o
Bonn, Bonn Ge many.
Email: linmei.shang@il .uni-bonn.de
Funding in o ma ion
Deu sche Fo schungsgemeinscha , G an /
Awa d Numbe : EXC- 2070- 390732324-
PhenoRob; Eu opean Commission, G an /
Awa d Numbe : 817566- MINDSTEP
Abs ac
Technological change co- de e mines ag i- en i onmen al
pe o mance and a m s uc u al ans o ma ion.
Meaning ul impac assessmen o ela ed policies can be
de i ed om a m- le el models ha a e ich in echnol-
ogy de ails and en i onmen al indica o s, in eg a ed wi h
agen - based models cap u ing dynamic a m in e ac ion.
Howe e , such in eg a ion aces conside able challenges
a ec ing model de elopmen , debugging and compu a-
ional demands in applica ion. Su oga e modelling using
deep lea ning echniques can acili a e such in eg a ion
o simula ions wi h b oad egional co e age. We de elop
su oga es o he a m model Fa mDyn using di e en a -
chi ec u es o neu al ne wo ks. Ou speci ically designed
e alua ion me ics allow p ac i ione s o assess ade- o s
among model i , in e ence ime and da a equi emen s.
All es ed neu al ne wo ks achie e a high i bu di e
subs an ially in in e ence ime. The Mul ilaye Pe cep on
shows almos op pe o mance in all c i e ia bu sa es
s ongly on in e ence ime compa ed o a Bi- di ec ional
Long Sho Te m Memo y.
KEYWORDS
agen - based model, deep lea ning, a m modelling, neu al ne wo ks,
su oga e model, upscaling
JEL CLASSIFICATION
C45, C63, Q12, Q18
236
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SHANG ET AL.
1 | INTRODUCTION
Modelling he impac s o ag i- en i onmen al policies inc easingly equi es accoun ing o
de ailed a m- le el decision- making, he e ogeneous local condi ions, and in e ac ion among
a me s. Policies ha a e ela i ely homogeneous ac oss egions (such as a i s and expo
subsidies a he EU le el o decoupled income suppo ) a e con inuously subs i u ed o
complemen ed wi h mo e a ge ed a m- le el policies— o example, he newly in oduced
eco- schemes o collec i e ag i- en i onmen al paymen s ha equi e coo dina ion and pa ic-
ipa ion o local communi ies (Kuh uss e al.,2016; Šum ada e al.,2022). De ailed a m- le el
models (Richa dson e al.,2014; Wee sink e al.,2002), usually implemen ed as op imisa ion
models, a e capable o ep esen ing indi idual decision- making wi h a ich ep esen a ion o
inpu choices, in es men s and en i onmen al indica o s. Howe e , hose a m- le el mod-
els usually do no accoun o in e ac ion among a me s, ma ke eedback, o en i onmen-
al eedback on la ge scales (Heckelei,2013; Shang e al.,2021). He e, agen - based models
(ABMs) (Gilbe ,2007) can be used o model endogenous ma ke eedback and o cap u e
he dynamic in e ac ion o he e ogeneous a ms (K emmydas e al.,2018; Mülle e al.,2020;
Rasch e al.,2017). Howe e , compu a ional demands limi he complexi y o a m decision-
making models wi hin an ABM o he numbe o agen s and hence he egional co e age o
hose models (B adhu s e al.,2016; Mu ay- Rus e al.,2014; Sun e al.,2016). In eg a ing
de ailed a m- le el models as indi idual decision- making models in o ABMs— while s ill co -
e ing a la ge egion— is desi able o policy analysis bu usually causes high compu a ional
cos s in applica ion, di icul ies in da a exchange, and challenges in model upda e/debugging.
We add ess his issue by aining and e alua ing compu a ionally e icien su oga es ha can
be in eg a ed in o ABMs in place o he o iginal a m models wi hou any ele an losses in
accu acy and de ail o model ou comes.
We demons a e he aining and e alua ion o su oga e models o he a m- le el model
Fa mDyn (B i z e al.,2016), which could be in eg a ed in o ABMs. To make he discussion
mo e conc e e, we conside he ABM Ag icul u al Policy Simula o (Ag iPoliS) (Appel &
Balmann,2019; Happe e al.,2006) as an example, bu su oga e models could equally be used
in o he ABMs. Fa mDyn is an economic simula ion ool ha is used ex- an e o assess ag icul-
u al policy e o ms and he adop ion o new echnologies. I simula es a m p oduc ion and
in es men decisions unde changes in p ices o inpu s/ou pu s, echnology and policy ins u-
men s o di e en a ming b anches in Ge many and o he coun ies (B i z e al.,2021). The
linkage o biophysical pa ame e s o highly de ailed a ming ac i i ies enables use s o assess
bo h economic and en i onmen al policies wi h a wide ange o social, economic and en i on-
men al indica o s a he a m le el. I has been applied, o example, o assess he impac o
he e ised Ge man e ilisa ion o dinance (Kuhn e al.,2020), he impac o changes in wa e
le els o pea soils on a m income (Poppe e al.,2021), he impac o Eu opean e ilise laws
on legume p oduc ion (Hein ichs e al.,2021) and he po en ial adop ion o a new pes icide ad-
di i e (Kuhn e al.,2022). The p o i - maximising solu ion o a a m is sol ed by Mixed- In ege
P og amming (MIP), which is ime- consuming when many a iables and cons ain s o di e -
en ypes a e in ol ed (Seidel & B i z,2019).
Ag iPoliS is a spa ial and dynamic ABM ha explici ly models a me s' in e ac ion on he
land ma ke . I has been used o s udy he impac on ag icul u al s uc u al change o di e en
policies, such as decoupling di ec paymen (Happe e al.,2008) and Ge many's biogas policy
(Appel e al.,2016). In Ag iPoliS, a me agen s maximise household income/p o i , which is
also sol ed by MIP. Compa ed o Fa mDyn, he MIP in Ag iPoliS is simple because i models
less de ailed echnology choices and aces ewe cons ain s (e.g., en i onmen al cons ain s).
Di ec in eg a ion o he MIP in Fa mDyn and Ag iPoliS o combine he s eng hs o bo h is
compu a ionally demanding and quickly becomes p ohibi i e as he spa ial co e age expands
(B adhu s e al.,2016; Hube e al.,2022; Sun e al.,2016). Besides, he wo models unning
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SURROGATE MODELLING USING DEEP LEARNING
oge he ende model upda es/debugging e y challenging. Howe e , combining he ad an-
ages o bo h ypes o models becomes inc easingly necessa y o ag i- en i onmen al policy
analysis (Hube e al.,2018).
Su oga e models, also known as me amodels o emula o s, may sol e his p oblem (Jiang
e al.,2020; Ra o e al.,2012). They app oxima e compu a ionally cos ly simula ion models
by mapping he ela ionship be ween inpu s and ou pu s while being much cheape o un.
The a ailabili y o highly lexible machine lea ning ools such as neu al ne wo ks (NNs)
(Good ellow e al.,2016) o e s he oppo uni y o build su oga es o complex and compu a-
ionally demanding simula ion models (Raza i,2021; S o m e al.,2020). In his way, a su o-
ga e model unc ions as a b idge be ween de ailed a m- le el models and la ge- scale ABMs o
e icien ly u ilise he ad an ages o bo h ypes o models. Al hough pa allel simula ion on a
high- pe o mance compu e (see An e al.,2021) can also sa e compu a ional ime, su oga e
models p o ide se e al ad an ages: (1) High- pe o mance compu ing is no always a ailable
o access migh be limi ed, which can be a bo leneck, pa icula ly du ing de elopmen ; (2)
Ha ing a compu a ionally mo e e icien su oga e model allows a la ge numbe o expe i-
men s o be pe o med in a sho ime wi h he same compu a ional esou ces; (3) Su oga e
models allow a mo e na u al sepa a ion o he de elopmen o he a m- le el model and he
ABM. This allows us o modula ise he in eg a ed modelling sys em, which can in he long un
simpli y model upda e/debugging and os e model eusabili y o he bene i o o he esea ch-
e s (B i z e al.,2021). I can also simpli y collabo a ion be ween di e en esea ch g oups in
e ms o so wa e licensing and da a access issues. Fo example, he a m- le el model migh be
un in a p op ie a y so wa e en i onmen ha migh no be a ailable o he g oup unning
he ABM, while he su oga e model is ained in Py hon wi h all pa s being open- access.
Su oga e modelling has been applied in a ious ields, such as wa e esou ce modelling
(Raza i e al.,2012), enginee ing (Jiang e al.,2020), wea he o ecas ing (Chen e al.,2020),
and ag icul u al economics (T oos e al.,2022). T oos e al.(2022) de elop di e en ypes o
su oga e models o app oxima e a a m- le el model using mul inomial- logis ic eg ession,
mul i a ia e adap i e eg ession splines, andom o es eg ession and ex eme g adien boos -
ing. Thei su oga e models cap u e he unde lying ela ionship be ween 22 inpu s (p ices and
model unce ain y pa ame e s) and 9 ou pu s (c op a eas). Howe e , o ou knowledge, he
applica ion o su oga e modelling using NNs in ag icul u al economics does no ye exis . In a
b oade sense, he e a e only wo s udies— Audsley e al.(2008) and Nguyen e al.(2019)— ha
use NNs o app oxima e a c op model and biogeochemical model o p edic c op yields and
soil o ganic ca bon, which a e u he used in economic models. Howe e , bo h s udies use a
classical ype o NN, mul ilaye pe cep on (MLP). To he bes o ou knowledge, su oga e
models o a de ailed a m op imisa ion model wi h di e en a chi ec u es o NNs is unex-
plo ed in ag icul u al economics. We de elop su oga e models o Fa mDyn as a i s s ep
such ha i can be in eg a ed in o ABMs like Ag iPoliS.
We see ou main con ibu ions. Fi s , we show i is possible o build well- i ed su oga es o
de ailed a m- le el models using NNs. Second, we sys ema ically compa e he pe o mances o
di e en a chi ec u es o NNs. Thi d, we de elop a se o e alua ion me ics o assess he qual-
i y o su oga e models. He e, we go beyond c i e ia such as
R2
o mean squa ed e o (MSE)
and de elop gene ic me ics ha can also be applied o e alua e o he su oga e models. They
help judge i he ained su oga e p o ides he equi ed accu acy o he in ended pu poses.
This is essen ial because di e en NN a chi ec u es de ia e subs an ially in in e ence ime
(i.e., he ime o make one p edic ion) wi h only mino di e ences in
R2
o MSE. Thus, mo e
de ailed and p ac ically ele an e alua ion me ics a e equi ed o judge i hose di e ences
in
R2
o MSE a e o p ac ical impo ance and jus i y he inc eased in e ence ime. Fou h, we
in es iga e he pe o mance o su oga e models gi en di e en amoun s o aining da a o
p o ide p ac ical guidance o modelle s. While i is possible o inc ease he amoun o da a by
unning he unde lying model delibe a ely, i is o en compu a ionally expensi e. Hence, o
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SHANG ET AL.
p ac ical pu poses, i is c ucial o de e mine how much da a is equi ed o di e en a chi ec-
u es o NNs o achie e he desi ed pe o mance on he de ined e alua ion me ics.
This es o he pape is o ganised as ollows. Sec ion2 e iews exis ing su oga e models o
iden i y he common a chi ec u es o NNs cu en ly used in he li e a u e. Sec ion3 in oduces
he o e all esea ch design. In Sec ion4, we analyse he esul s and assess he pe o mance o
NNs gi en di e en amoun s o aining da a. Sec ion5 p o ides a concep ual discussion on
using su oga e models in ABMs o ag icul u al policy simula ion. The las sec ion concludes
and poin s ou di ec ions o u u e esea ch.
2 | NNS AS SURROGATE MODELS IN THE LITERATURE
Su oga e models in he li e a u e a e based on a la ge a ie y o model ypes, including poly-
nomial eg ession (Hussain e al.,2002), adial basis unc ions (Amouzga & S ömbe g,2017),
k iging (Kleijnen,2009), Gaussian p ocesses (Picheny,2015), suppo ec o machines (Xiang
e al.,2017), gene ic p og amming (Fallah- Mehdipou e al.,2013), Bayesian ne wo ks (G ube
e al.,2013) and NNs (Sun & Wang,2019). Th oughou his pape , we ocus on NNs as hey b ing
new p omise o su oga e models ha equi e lowe compu a ional cos (Chen e al.,2021).
This sec ion in oduces basic concep s o NNs and iden i ies he common a chi ec u es o
NNs used as su oga es in he li e a u e. No e ha he app oach o eplacing agen s' decision-
making wi h NN- based su oga e models is di e en om he app oach ha uses NNs as
unde lying s uc u e in ABM (e.g., Jäge ,2021).
2.1 | Basic concep s o NNs
NNs a e capable o ep esen ing highly non- linea ela ionships and a e well placed o deal
wi h high dimensions in he inpu and he ou pu space. Figu e1a depic s he mos commonly
used a chi ec u e o NN: MLP. I consis s o an inpu laye , an ou pu laye , and a leas one
hidden laye be ween he wo. Each laye con ains a ce ain numbe o neu ons. Like a bio-
logical neu on, an a i icial neu on p ocesses he in o ma ion om he inpu s in he p e ious
laye and ans e s he signal o he nex neu on, as shown in Figu e1b. An a i icial neu on
pe o ms wo s eps o compu a ion. Fi s , a weigh ed sum o all inpu s is compu ed as shown
in Equa ion(1):
FIGURE 1 The a chi ec u e o an MLP (a) and an a i icial neu on (b).
xi
is he alue o an inpu neu on,

yi
is
he p edic ion o an ou pu neu on,
wi
is he weigh o a neu on,
b
is he bias,
z
is he ou pu o he weigh ed sum,
and
(
z)
ep esen s he ac i a ion unc ion. Sou ce: Based on Good ellow e al.(2016).
14779552, 2024, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1111/1477-9552.12543 by Coch ane Ge many, Wiley Online Lib a y on [14/02/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

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SURROGATE MODELLING USING DEEP LEARNING
whe e
wi
is he weigh o he inpu neu on
xi
,
b
is he bias, m is he numbe o inpu neu ons, and z
is he weigh ed sum.
Second, he weigh ed sum will be ans e ed by an ac i a ion unc ion (
(z)
). Typically,
ac i a ion unc ions a e non- linea . Fo example, he Rec i ied Linea Uni (ReLU) e u ns he
alue ha is equal o he inpu i i is posi i e, and i e u ns ze o o he wise.1
Weigh s and biases a e called ‘pa ame e s’ o an NN. T aining an NN like MLP inds he
op imal pa ame e s o minimise he loss unc ion, ha is, a unc ion ha measu es he di -
e ence be ween he p edic ed ou pu s and he simula ed ou pu s (e.g., he MSE loss). This
p ocess is usually done i e a i ely h ough backp opaga ion algo i hms ha compu e he g a-
dien o he loss unc ion wi h espec o he weigh s and biases (Rumelha e al.,1986). The
g adien s a e hen used by an op imisa ion algo i hm (an op imise ) o upda e he pa ame e s.
T aining he NN wi h all he aining da a o one cycle is called one ‘epoch’. Usually, NNs a e
ained o mul iple epochs. Wi hin one epoch, he aining da ase can be di ided in o mini-
ba ches, which will be passed h ough o he NN a one ime. The numbe o da a poin s ha
a mini- ba ch con ains is called he ‘mini- ba ch size’.
While pa ame e s can be es ima ed by algo i hms om he aining da a, ‘hype pa ame-
e s’ canno be es ima ed om he da a and a e usually se manually by he modelle be o e
aining. NNs ha e a ious hype pa ame e s (like he numbe o laye s and neu ons). They
may in e ac wi h each o he in non- linea ways. Hype pa ame e uning is a p ocedu e o
inding he op imal hype pa ame e s o an NN (o o he machine lea ning models), in o-
duced in de ail in Sec ion3.
2.2 | Di e en a chi ec u es o NNs used in su oga e modelling
Mul ilaye pe cep ons ha e been widely used as su oga es in di e se disciplines (Roman
e al.,2020). I has been shown ha an MLP o one hidden laye (i.e., a shallow NN) wi h
an adequa e numbe o neu ons can be ained o app oxima e any measu able unc ion
o any desi ed deg ee o accu acy (Ho nik e al.,1989). As a esul , s udies using shallow
MLPs a e common in su oga e modelling. Fo example, Ca ne ale e al.(2012) use a one-
hidden- laye MLP o lea n he ela ionship be ween emissions and ai quali y indices. In
he e iew by Raza i e al.(2012) o su oga e models in wa e esou ce modelling, 13 ou
o 14 pape s used shallow NNs. Howe e , deep NNs (i.e., wi h mo e han one hidden laye )
migh equi e ewe neu ons o cap u e a simila le el o complexi y and hus a e also ap-
plied as su oga es. Fo ins ance, Liong e al.(2001) use an NN wi h h ee hidden laye s o
mimic a hyd ological model.
The second common ype o NNs used as su oga e models is con olu ional neu al ne -
wo ks (CNNs) (LeCun e al.,1990), o iginally designed o image da a. In con as o MLPs
whe e he neu ons in one laye a e connec ed o all neu ons o he p e ious laye , CNNs use
so- called ‘con olu ion ke nels’ ha slide ac oss he inpu . Each neu on hus depends only on
a local neighbou hood o neu ons and no on all neu ons o he p e ious laye . CNNs a e also
p omising in handling ime- se ies da a (Fawaz e al.,2019). Al hough deepe CNNs migh be
able o cap u e mo e complex ela ionships, classical CNNs do no pe o m well as hey g ow
deepe due o he p oblem o anishing g adien (i.e., he g adien s o he loss unc ion ap-
p oach ze o, making NNs ha d o ain) (Bengio e al.,1994). To o e come his issue, esidual
ne wo ks (ResNe s) (He e al.,2016) allow ‘skip connec ions’ o enable he aining o deepe
ne wo ks. An addi ional skip connec ion skips mul iple laye s o a neu al ne wo k such ha
he ou pu o one laye is no only ed o he nex laye bu also o he a ge laye o he skip
(1)
z
=
∑m
i=1
wixi+
b
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SHANG ET AL.
connec ion. Webe e al.(2019) ind ResNe s pe o m be e han classical CNNs in su oga e
modelling o clima e o ecas s.
The hi d common ype o NNs used is ecu en neu al ne wo ks (RNNs) (Elman,1990;
Rumelha e al.,1986; We bos,1988), designed o sequence p edic ion asks, such as speech
ecogni ion (G a es e al.,2013) and ime se ies modelling (Hsu,2017). RNNs a e sui able o
p ocessing sequen ial da a since he ecu en laye s eed he ou pu o he laye back o he
laye i sel such ha he cu en s a e o a laye depends on he cu en inpu o he sequence as
well as on he p e ious s a es o he laye . Howe e , as he leng h o inpu s inc eases, long- e m
dependencies a e di icul o cap u e by classical RNNs (Ma hon e al.,2013). Long sho -
e m memo y (LSTM) (Hoch ei e & Schmidhube ,1997) is a special RNN, capable o lea n-
ing long- e m dependencies. Rahmani e al.(2021) ha e de eloped LSTMs as su oga es o a
p ocess- based model o p edic s eam wa e empe a u e. LSTMs ha e been used o p edic
c op yields (e.g. Sun e al.,2019; Tian e al.,2021), bu o ou knowledge, hey a e no ye applied
as su oga es o ag icul u al models. The BiLSTM (bidi ec ional long sho - e m memo y)
(G a es e al.,2005) is an ex ension o LSTM. I lea ns he sequence and he e e sed sequence
o he inpu s. Alibabaei e al.(2021) use a BiLSTM o model e apo anspi a ion and soil wa e
con en in i iga ion scheduling. RNNs a e also help ul o non- sequen ial da a. Fo example,
Chop a e al.(2017) ain an RNN wi h non- sequen ial da a o p edic whe he a pa ien would
be eadmi ed o he hospi al.
Al hough MLPs ha e been applied as su oga es by Audsley e al.(2008) and Nguyen
e al.(2019) (see Sec ion1), and LSTMs ha e been used o p edic c op yields as men ioned
abo e, using NNs o di e en a chi ec u es as su oga es is unexplo ed in ag icul u al mod-
els. Besides, no NN applica ions o app oxima e economic a m models a e known o us.
Gi en hese esea ch gaps, we employ he ou di e en a chi ec u es o NNs including
MLP, ResNe , LSTM and BiLSTM o de elop su oga es o he de ailed a m- le el model
Fa mDyn.
3 | METHOD AND DATA
Ou esea ch design is shown in Figu e2. Fi s , om he unde lying a m model Fa mDyn, we
gene a e he da a ha will be used o aining NNs. This in ol es de ining he inpu s/ou pu s
o he a m model, gene a ing da a, and some da a p epa a ion s eps. Second, o each o he
ou NN a chi ec u es, we de ine h ee di e en implemen a ions ha a y in dep h (i.e., he
numbe o laye s). This esul s in 12 a ian s o dep h, o which we op imise he emaining
hype pa ame e s. The loss unc ion used o ain NNs is he MSE loss. I should be no ed
ha minimising MSE is by cons uc ion equi alen o maximising
R2
(see Equa ion2). We
hen selec one bes model in e ms o
R2
om each a ian o dep h (in o al 12 bes models)
and compa e hei in e ence ime. Thi d, om each NN a chi ec u e, we selec he bes model
wi h he mos p omising hype pa ame e s and inspec model pe o mance in g ea e de ail.
Speci ically, we examine model pe o mance ac oss a ying amoun s o aining da a by con-
side ing a se o e alua ion me ics. The de ails o hese h ee s eps a e desc ibed below.
3.1 | The unde lying model and da a gene a ion
3.1.1 | De ine inpu s/ou pu s o he a m model
The bio- economic a m- le el model Fa mDyn co e s a wide ange o a m b anches, such as
a able, dai y, bee ca le, pig a ening and biogas. We ocus on he a able a ming b anch.
Howe e , as a obus ness check o he su oga e modelling pipeline, AppendixS1 p esen s
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SURROGATE MODELLING USING DEEP LEARNING
an addi ional smalle expe imen o dai y a ms in Fa mDyn. In gene al, a able a ms in
Fa mDyn make managemen decisions by maximising he ne p o i using MIP. In he e sion
we use o his pape , he p og amming p oblem is cons uc ed wi h se e al modules, wi h
each module dealing wi h a speci ic aspec o a a m. Fo mo e de ails, see he documen a ion
o Fa mDyn: h ps:// a md yn.gi hub.io/docum en a ion/.
1. Economic module: his de ines he economic aspec s o he a m- le el model, including
objec i e unc ion (ne p o i maximisa ion), cash low s uc u e, income ax calcula ion,
p emium paymen s, sales and p oduc ion le els, a iable cos s uc u e, and in es men
cos s.
2. Gene al c opping module: his op imises he c opping decisions subjec o land a ailabili y,
yields, maximal c op o a ional sha es, c op p ices, machine y and e ilise needs, and o he
a iable cos s o c ops.
3. Labou module: his op imises labou alloca ion o di e en de ailed on- and o - a m ac-
i i ies wi h a mon hly esolu ion.
4. En i onmen al accoun ing: his quan i ies a m- le el me hane (CH4), ammonia (NH3), ni-
ous dioxide (N2O), ni ogen oxides (NOx) and elemen al ni ogen (N2), as well as pa icu-
la e ma e o ma ion (PM10 and PM2.5).
5. Fe ilisa ion o dinance: his adds he e ilisa ion cons ain s based on he Ge man imple-
men a ion o he Ni a es Di ec i e, including nu ien balance es ic ions, h eshold o
o ganic ni ogen applica ion quan i ies, es ic ion o e ilise applica ion in au umn, and
o he s.
FIGURE 2 The o e all esea ch design o his s udy.
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6. G eening: his cap u es he g eening equi emen s o he Common Ag icul u al Policy o
he Eu opean Union (CAP) as applicable since 2013. I in eg a es he key measu es o c op
di e si ica ion and ecological ocus a eas (mainly legumes) in o Fa mDyn.
To build a su oga e model o Fa mDyn, i is necessa y o de ine he model in e ace clea ly.
This means we need o de ine wha inpu a iables we pass o he model and wha ou pu a i-
ables we aim o ob ain. In ou case, he su oga e model akes he same inpu s and p oduces
he same ou pu s as he unde lying model Fa mDyn. The e o e, de ining he inpu s/ou pu s o
Fa mDyn will echnically de ine he inpu s/ou pu s o he su oga e model.
Table1 summa ises he inpu s and ou pu s o a able a ms in Fa mDyn. They include a i-
ables abou c ops, a ming inpu s, machine y, a m endowmen , en i onmen al indica o s,
and a m accoun ing. C ops included in he model a e win e whea , win e ba ley, win e
apeseed, summe ce eal, maize and suga bee . The a ming inpu s include diesel, e ilise
(u ea- ammonium ni a e, phospho us and po assium), seed, lime, he bicide, ungicide, insec-
icide, g ow h con ol, wa e , and hail insu ance. In o al, he e a e 77 inpu s and 248 ou -
pu s. The e a e many cons an pa ame e s in Fa mDyn, bu we exclude hem he e since he
su oga e model should be able o lea n he unde lying cons an pa ame e s ha e lec he
ela ionship be ween inpu s and ou pu s. The de ailed lis s o inpu s and ou pu s can be ound
in AppendixS1.
We de elop a su oga e model ha p edic s 248 ou pu s simul aneously ins ead o building
one sepa a e model o each ou pu . The ad an ages o his app oach a e: (1) i scales be e
wi h he numbe o ou pu s and is less ime- consuming han aining many sepa a e models;
and (2) i is easie o in eg a e only one su oga e model in o he u u e ABM han many small
ones. Howe e , aining sepa a e models o each ou pu makes i easie o obse e he loss
unc ion o each ou pu , hus i could be easie o imp o e he model accu acy o some pa -
icula ou pu s. None heless, when aining a neu al ne wo k wi h mul iple ou pu s, one can
weigh hem di e en ly, hen he loss unc ion will eac mo e o hose ‘mo e impo an ’ ou -
pu s. In his pape , we ea all ou pu s equally since we do no a ge any speci ic applica ion
he e.
3.1.2 | Da a gene a ion and p epa a ion
The ini ial a m da a is gene a ed om Fa mDyn by La in Hype cube Sampling (LHS) (McKay
e al.,1979). LHS independen ly s a i ies each inpu dimension in o N equal in e als, whe e N
is he numbe o da a poin s. Fo a gi en dimension, i gene a es one da a poin in each in e al
and andomly combines his wi h he selec ed in e al o he o he dimensions. LHS p o ides
ou comes om a uni o m dis ibu ion o he da a wi hin he design space (Tyan & Lee,2019).
The op imal amoun o da a used o ain a su oga e model depends on he complexi y o
he p oblem and he compu a ional budge a ailable. Since NNs need la ge da ase s o p ob-
lems wi h high dimensionali y, we gene a ed as many da a poin s as possible gi en ou ime
budge . Wi h 10,000 model ou comes (i.e., obse a ions) each ime, he da a gene a ion p ocess
an 17 imes and p oduced 163,480 da a poin s ( aking abou 45 h) because Fa mDyn did no
success ully sol e o some inpu d aws due o implausible inpu combina ions.
The whole da ase is hen andomly spli in o wo subse s including he aining se (90%)
and he es se (10%), ha ing 147,132 and 16,348 obse a ions, espec i ely. The aining se is
used o ain he model, and a es se is solely used o assess he model. Du ing he aining
p ocess, 10% o he aining se is used as a alida ion se o moni o he models' pe o mance
on unseen da a o a oid o e i ing, meaning he ne wo k lea ns oo much in o ma ion ha
is speci ic o he aining da a and does no gene alise o o he da ase s. The alida ion se is
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SURROGATE MODELLING USING DEEP LEARNING
The a e age
APE ela ionship
o he abo e- men ioned wo g oups o a iables is calcula ed a
he end.
3. Accu acy in cap u ing co ne solu ions
Ano he impo an aspec o he applica ion o su oga e models is i s abili y o cap u e
co ne solu ions. These a e special solu ions o an op imisa ion p oblem in which he quan i y
o one o he a gumen s in he objec i e unc ion is ze o (Debe in,2012). In a able a ming,
examples o co ne solu ions a e an a ailable echnology ha is no chosen o a pa icula c op
ha is no p oduced. A p e ious s udy has shown ha cap u ing co ne solu ions is usually
challenging o su oga e models (Seidel & B i z,2019). The abili y o he model o cap u e co -
ne solu ions is di icul o assess om
R2
. I would be in e es ing o see i he su oga e model
is able o cap u e co ne solu ions, ha is, i i a leas ge s he a me s' basic c op choices
co ec wi hou conside ing he le el. This dimension becomes pa icula ly ele an i a me s'
choices a e he ocus o he analysis in applying su oga e models, o example when simula ing
a me s' echnology adop ion decisions.
Fo example, we measu e NNs' abili y o cap u e co ne solu ions o a me s' c op choices.
Fo a c op c, we i s ans o m i s simula ed and p edic ed p oduc ion le els o each obse a-
ion in o bina y: 0 (i no p oduced4) and 1 (i p oduced). Then, we coun he numbe o a ms
whose decisions a e co ec ly p edic ed. The accu acy in cap u ing co ne solu ions o c op c
is calcula ed wi h Equa ion(7):
whe e
ac
is he numbe o obse a ions whose decision on c op c is co ec ly p edic ed, and N is
he numbe o obse a ions in he es se .
The a e age accu acy in cap u ing co ne solu ions ac oss all c ops is calcula ed wi h
Equa ion(8):
whe e C is he numbe o c op ypes (C = 6 in his s udy).
4. Accu acy in holding cons ain s
Indi idual a m op imisa ion models simula e a me s' choices o maximise an ou pu subjec
o a se o cons ain s (e.g., land/labou endowmen ). When employing a su oga e model o such
an indi idual a m model, i is c ucial ha hose cons ain s hold. Fo example, he sum o he
plan ed a eas o all a m c ops canno exceed he a m size i en ing land is impossible. F om an
economic modelling poin o iew, a smalle iola ion o hese cons ain s by he su oga e model
is o en mo e p oblema ic han a la ge de ia ion om he unde lying model beha iou wi hin
he easible solu ion space (e.g., some unde u ilisa ion o a esou ce).
R2
does no cap u e his, as
i does no dis inguish be ween easible and in easible solu ion space gi en by he cons ain s o
he unde lying model. The e o e, a dedica ed measu e o how well he p edic ion o he su oga e
model obeys he cons ain s is wa an ed.
As an example, we measu e NNs' accu acy in holding cons ain s o a m size wi h Equa ion(9):
whe e
acons ain
is he numbe o obse a ions whose cons ain s o a m size a e no iola ed, and
N is he numbe o obse a ions in he es se .
(7)
A
c=
1
N
a
c
(8)
Accu acy
co ne =
1
C∑
A
c
(9)
Accu acy
cons ain =
1
N
a
cons ain
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3.3.3 | T aining wi h di e en amoun s o da a
To in es iga e he impac o he amoun o aining da a on he pe o mance o he su oga e
model, we choose he bes model wi h he mos p omising hype pa ame e s om each NN
a chi ec u e and ain hem wi h a ying amoun s o aining da a. We spli he o iginal ain-
ing se (Sec ion3.1.2) in o sizes o {1000, 5000, 10,000, 50,000, 100,000, 147,1325}. The es se is
he same as be o e, con aining 16,348 da a poin s, bu i is no malised acco ding o he scale
o each aining se . To a oid luc ua ions, we a e age he pe o mances o i e models ained
wi h he same da a using di e en andom seeds o each a chi ec u e o NN and o each size
o aining se .
4 | RESULTS AND DISCUSSION
4.1 | The bes models and hei in e ence ime
We selec he 12 bes models in o al ( h ee a ian s o dep h om each a chi ec u e) in e ms
o
R2
on he es se . Table4 shows he a chi ec u e o he selec ed NNs. As can be seen,
BiLSTM3 (BiLSTM wi h h ee hidden laye s) has he highes
R2
o 0.99, while ResNe 18 has
he lowes
R2
o 0.93. This shows NNs can cap u e he a iance in he da a e y well. In e ms
o
R2
, we obse e ha BiLSTMs and LSTMs pe o m be e han MLPs and ResNe s. RNNs,
al hough designed o sequen ial da a, can also adap o non- sequen ial da a. AppendixS1
p o ides he de ailed sca e plo s o he p edic ions o BiLSTM3 ( he model wi h he highes
R2
) and simula ed esul s o a ew ou pu s ha a e usually impo an in applica ions.
As shown in Figu e3, he in e ence ime o di e en NNs di e s subs an ially. MLPs a e
he as es in p edic ing, whe eas LSTMs and BiLSTMs a e much slowe , e lec ing he la ge
numbe o pa ame e s han MLPs (see Table4). Fa mDyn akes 5.40 s o gene a e one da a
poin on a e age. In compa ison, he MLP3 (MLP wi h h ee hidden laye s) (
R2
= 0.95) needs
0.000026 s o p edic one da a poin being abou 207,000 imes as e han Fa mDyn, and he
BiLSTM3 (
R2
= 0.99) akes 0.021 s, being 257 imes as e . Whe he his speed is sa is ying de-
pends on he ime budge o u u e applica ions.
4.2 | Model pe o mance and impac o he amoun o aining da a
Acco ding o Table4, we selec he ou bes model speci ica ions in e ms o
R2
o expe imen
wi h di e en sizes o aining se as desc ibed in Sec ion3.3.3. They a e MLP wi h 2 hidden
laye s (MLP2), ResNe wi h 50 laye s (ResNe 50), LSTM wi h 3 hidden laye s (LSTM3), and
BiLSTM wi h 3 hidden laye s (BiLSTM3). In he ollowing, we e e o hem as MLP, ResNe ,
LSTM and BiLSTM wi hou epea ing he numbe o laye s.
4.2.1 | Goodness o i
Figu e4a,b show he change o
R2
and RMSE (calcula ed using he no malised da a) o he
selec ed NNs wi h a ying amoun s o aining da a. Wi h a aining se o 1000 obse a ions,
BiLSTM and MLP can achie e an a e age
R2
o 0.8, whe eas LSTM can only achie e a ound
0.55. Fo ResNe , 1000 obse a ions o aining a e insu icien o con e ge because he
R2
o ResNe ained wi h his amoun o da a is nega i e (no shown in he igu e).6 As he size
o aining se inc eases om 1000 o 5000, we see a s eep inc ease in
R2
o all ou ypes o
models. Wi h 50,000 da a poin s o aining, BiLSTM and MLP can al eady achie e a
R2
o
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SURROGATE MODELLING USING DEEP LEARNING
TABLE 4 The a chi ec u es o he 12 selec ed models based on R2 on he es se .
Numbe o hidden
laye s
Numbe o neu ons in
each hidden laye
Numbe o il e s in
he second s age
Lea ning
a e
Mini- ba ch
size Op imise
Numbe o
pa ame e s R2
MLP1 1128 /0.001 32 RMSp op 41,976 0.94
MLP2 264, 512 /0.0003 32 Adam 165,496 0.96
MLP3 3 128, 32, 256 / 0.0003 32 Adam 86,296 0.95
ResNe 18 18 /32 0.001 128 Adam 1,119,960 0.93
ResNe 34 34 / 8 0.0003 32 Adam 171,648 0.94
ResNe 50 50 /16 0.001 64 Adam 1,666,648 0.94
LSTM1 1 256 / 0.001 32 Adam 327,928 0.97
LSTM2 2128, 64 /0.001 32 Adam 132,088 0.97
LSTM3 3 32, 128, 1024 / 0.001 32 Adam 5,063,672 0.98
BiLSTM1 12048 /0.001 32 Adamax 34,603,256 0.98
BiLSTM2 2 32, 256 / 0.001 32 Adamax 793,336 0.98
BiLSTM3 332, 128, 512 /0.001 32 Adamax 3,610,360 0.99
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SHANG ET AL.
a ound 0.95. In e es ingly, wi h 100,000 da a poin s, all models excep o LSTM a e al eady
close o hei maximum pe o mance le el, whe e addi ional da a is o li le bene i .
4.2.2 | Consis ency o bi a ia e ela ionships
Figu e4c shows he measu e o he abili y o cap u e he ela ionships o wo g oups o a i-
ables as men ioned abo e. Wi h mo e and mo e aining da a, he
APE ela ionship
o BiLSTM
goes down s eadily, achie ing an APE o 0.70% wi h 100,000 obse a ions. In compa ison,
MLP can also each a simila le el o accu acy bu wi h luc ua ions when he size o ain-
ing se is smalle . BiLSTM, MLP and LSTM all achie ed he bes pe o mance in cap u ing
he ela ionships wi h 100,000 obse a ions, while ResNe has a much highe le el o e o o
8.35% gi en he same amoun o aining da a.
4.2.3 | Accu acy in cap u ing co ne solu ions
Figu e4d shows he accu acy in cap u ing co ne solu ions o c op choices (
Accu acyco ne
)
o each NN a chi ec u e ained wi h di e en amoun s o da a. Wi h 10,000 da a poin s o
aining, BiLSTM can achie e accu acy nea o 100% in cap u ing he co ne solu ions o c op
choices. Once he size o aining se exceeds 50,000, he accu acy does no inc ease much o
mos models excep o LSTM. We can also see ha MLP is as good as BiLSTM in cap u ing
co ne solu ions a and beyond 50,000 da a poin s.
4.2.4 | Accu acy in holding cons ain s
Figu e4e shows he accu acy o NNs in holding cons ain s o a m size (
Accu acycons ain
). Wi h
a smalle aining se (less han 20,000 da a poin s), MLP ou pe o ms BiLSTM wi h an accu acy
o 0.98, bu BiLSTM domina es once he size o aining se eaches 50,000. Fu he mo e, he
FIGURE 3 In e ence ime pe da a poin o each NN.
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SURROGATE MODELLING USING DEEP LEARNING
accu acy o BiLSTM in holding he cons ain s is e y close o 100%, gi en 50,000 da a poin s.
A e his poin , adding mo e da a poin s does no imp o e he pe o mance o BiLSTM.
Figu e4 shows he o al sco e o each NN, which is calcula ed by simple addi ion and sub ac-
ion o all c i e ia (
To al sco e
=R
2
−RMSE −APE
ela ionship
+Accu acy
co ne
+Accu acy
cons ain
)
because hey all we e chosen o be in he ange o 0 and 1 in his s udy. As can be seen, inc easing
he size o aining se om 1000 o 50,000 signi ican ly imp o es he pe o mance o all ypes o
models. Once he size o aining se eaches 100,000, adding mo e obse a ions o he aining
p ocess does no necessa ily imp o e he pe o mance o su oga e models. Thus, in ou case, a
FIGURE 4 Pe o mance o di e en a chi ec u es o NNs gi en di e en sizes o aining se .
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size o aining se be ween 50,000 o 100,000 should be su icien o de elop su oga e models
ha pe o m well conce ning all ou e alua ion me ics. In e ms o model p e e abili y, BiLSTM
almos always domina es o e o he ypes o NNs gi en di e en amoun o aining da a bu has
a close compe i o — MLP. Conside ing he in e ence ime o he ained model, MLP may be he
go- o model in many su oga e model applica ions ha equi e a la ge numbe o model uns.
The su oga e models de eloped by T oos e al.(2022) cap u e he unde lying ela ionship
be ween 22 inpu s (p ices and model unce ain y pa ame e s) and 9 ou pu s (c op a eas). When
he size o he aining sample is below 1000 da a poin s, he pe o mance o he su oga e
models inc eases he mos . Wi h many mo e inpu and ou pu a iables in ou case, ain-
ing su oga e models equi es mo e da a. In e ms o in e ence ime, deep lea ning me hods
used in ou pape a e 257– 207,000 imes as e han he de ailed a m- le el model Fa mDyn,
whe eas he su oga e models in T oos e al.(2022) a e 1800– 60,000 imes as e han hei
unde lying a m model.
5 | SURROGATE MODELS FOR AGRICULTURAL POLICY
SIMULATION IN AN ABM: A CONCEPTUAL DISCUSSION
The p e ious sec ions show ha de eloping a su oga e model o a a m- le el model like
Fa mDyn is possible and hey p o ide p ac ical guidance o do so. He e, we discuss some im-
plica ions o how such a su oga e model can be used o ag icul u al policy simula ion in an
ABM, as well as u he a enues opened up by i . Addi ionally, we commen on he challenges
and po en ial downsides when using su oga e models.
In eg a ing a su oga e model o a a m- le el model in o an ABM makes i possible o ep e-
sen he decision- making mechanism o agen s (i.e., he beha iou o he unde lying indi idual
a m- le el model) wi h he su oga e model, whe eas he landscape o he selec ed egion and
in e ac ion ules among agen s a e de e mined by he ABM. P io o any simula ion, he ABM
ini ialises he a m popula ion o he selec ed esea ch egion. This migh include de ining he
ypes o a ms and he o iginal numbe o a ms ha belong o each a m ype, which e lec s
he cha ac e is ics o he a m popula ion in he esea ch egion. In he case o Ag iPoliS, a ms
a e ini ialised and di e en ia ed om each o he in e ms o loca ion, a m size, equi y, a ail-
abili y o labou , exis ing capaci y o machine y, age o he a m ope a o , and so on. Some
o hese a m cha ac e is ics se e as an inpu o he su oga e model o de e mining agen s'
beha iou . The ABM keeps ack o he ac ions o each agen and hei in e ac ions and up-
da es a m cha ac e is ics o he nex pe iod acco dingly. Fo example, i a a m has acqui ed
addi ional land o new in es men in one pe iod, i needs o be accoun ed o in he nex pe iod.
As one o he main oppo uni ies o using a su oga e model o couple complex a m- le el
models, such as Fa mDyn, wi h ABMs, such as Ag iPoliS, we conside he possibili ies o
simula e ag i- en i onmen al policy impac s. One o he s eng hs o Fa mDyn is he compa-
ably ich ep esen a ions o bio- physical p ocesses, a m echnologies and a m managemen
decisions. Fo example, Fa mDyn cap u es he whole ni ogen low (bio- physical p ocesses)
on he a m, which can be al e ed using low- emission manu e echniques ( a m echnology) o
expo ing on- a m manu e ( a m managemen decision). Fa mDyn also allows assessmen o
a wide ange o en i onmen al indica o s (e.g., ni ogen balances). This enables us o simula e
and assess ou comes o policies ha limi e ilise use in ce ain loca ions, e lec ing on he
one hand managemen decisions by indi idual a me s h ough Fa mDyn, and on he o he
hand accoun ing o (spa ially explici ) in e ac ions be ween a ms on he land ma ke simu-
la ed in Ag iPoliS. E en ually, he connec ion be ween bo h models allows us o assess spa ial
en i onmen al policy e ec s based on Fa mDyn's en i onmen al indica o s.
I is impo an o no e ha while su oga e models o e new possibili ies in coupling complex
indi idual a m- le el model wi h ABMs, i is s ill bound by he capabili ies o each indi idual
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model. Fo example, he cu en e sion o Fa mDyn, which is conside ed in his pape , does no
allow a ms o swi ch be ween ypes, o example u ning om an a able a m o a dai y a m.
Hence, his capabili y is also no included in he su oga e model. In AppendixS1, we p esen
an addi ional su oga e modelling expe imen which es s he applicabili y o he app oach on
a dai y a m ype. I a swi ch be ween a m ypes is equi ed om he ABM pe spec i e, ei he
Fa mDyn needs o be ex ended by his ea u e, o a su oga e model needs o be ained on da a
o mul iple a m ypes, which in e nally can de e mine he esul ing a m ype.
Beyond su oga e models o ABMs, an en i ely di e en po en ial use o su oga e mod-
els is o mo e e icien calib a ion o he unde lying model (S o m e al.,2020). Assume ha
we wan o calib a e Fa mDyn o some empi ically obse ed c op sha es, ha is, we wan o
minimise he di e ence be ween he obse ed c op sha es (calib a ion a ge ) and he c op
sha e ou pu o Fa mDyn. Fo he calib a ion case wi h Fa mDyn, we could conside yields as
calib a ion pa ame e s; howe e , o he echnology pa ame e s o p ices a e also concei able
(see B i z,2021 o a de ailed desc ip ion o his calib a ion p ocedu e). In he case o an NN-
based su oga e model, one would hen use yields as a ying inpu a iables in addi ion o he
mo e gene al inpu a iables such as a m endowmen s. In he nex s ep, he su oga e model
can lea n new inpu /ou pu ela ionships unde di e en yield le els. Fo he calib a ion in
Fa mDyn, we can hen use he ained su oga e model o ind he yield le el ha minimises
he di e ence be ween he Fa mDyn ou pu and he de ined calib a ion a ge , which is in ou
example he c op sha es. The po en ial ad an age o using he NN- based su oga e model o
calib a ion is ha g adien s (i.e., how ou pu s change in esponse o changes in inpu s) can be
calcula ed analy ically and in a highly e icien manne . On he con a y, o he unde lying
model, g adien s need o be calcula ed nume ically, which is compu a ionally expensi e. In
heo y, his idea can be ex ended o calib a e he ABM coupled wi h a su oga e model. In his
case, i would equi e building a su oga e model o he en i e ABM. The su oga e model
lea ns he ela ionship be ween he a ying inpu s (including pa ame e s o be calib a ed and
he inpu a iables o he ABM) and he ABM ou pu s. Pa ame e s o be calib a ed could be,
o example, one ha speci ies in e ac ion beha iou on he land ma ke . Simila o he cali-
b a ion o a a m- le el model using su oga e models, he ABM pa ame e s can be e icien ly
calib a ed acco ding o he g adien s.
Despi e he bene i s o su oga e modelling, we mus be awa e o i s limi a ions. Fi s , we
need o conside ha al hough su oga e models hemsel es a e compu a ionally e icien ,
aining su oga e models, especially hype pa ame e uning, is ime- consuming and equi es
conside able compu a ional esou ces (T oos e al.,2022). Second, deep- lea ning- based su -
oga e models a e es ic ed in hei alidi y o he ange o inpu alues in he aining da a.
This means ha once he anges o inpu da a a e ex ended, su oga e models mus be e-
ained. Re aining migh also be necessa y each ime he unde lying model is upda ed, ei he
o conside new ea u es o o esol e bugs, o i he model needs o be adjus ed o a new s udy
o esea ch ques ion. This equen need o e aining migh coun e ac he ad an age o e-
usabili y o su oga e models. Howe e , u u e esea ch can o e come he di icul y by au o-
ma ing he aining p ocess as a as possible. I is also impo an o conside ha i migh no
be necessa y o epea he en i e hype pa ame e sea ch p ocess, as long as he undamen al
complexi y o he model is no changed subs an ially. This makes e aining subs an ially less
cos ly and au oma ion mo e easible.
6 | CONCLUSIONS
We in es iga e he pe o mance o NNs o di e en a chi ec u es in app oxima ing he beha -
iou o a de ailed a m- le el model Fa mDyn. We compa e he pe o mances o ou a chi ec-
u es o NNs (MLP, ResNe , LSTM and BiLSTM), conside ing 12 di e en implemen a ions
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SHANG ET AL.
in e ms o model dep h. The ained NNs a e supposed o accu a ely map he ela ionship
be ween 77 inpu a iables and 248 ou pu a iables o he a m model. The high goodness o
i o he selec ed su oga e models shows ha NNs can explain mos o he a ia ion in he
ou pu a iables. The BiLSTM wi h h ee hidden laye s achie es an a e age
R2
o 0.99 ac oss
all ou pu a iables, whe eas he lowes a e age
R2
is 0.93 by ResNe wi h 18 laye s. BiLSTM
and LSTM achie e be e pe o mance han o he ypes o NNs, al hough hey a e o iginally
designed o handle sequen ial da a. In e ms o in e ence ime, all ained NNs a e much as e
han Fa mDyn. MLPs a e abou 207,000 imes as e , and he bes pe o ming BiLSTM e-
ga ding
R2
is s ill 257 imes as e .
We also p o ide gene ic e alua ion me ics o assess he pe o mance o su oga e mod-
els, which can o e u u e modelle s addi ional help in selec ing su oga e models in ap-
plied modelling. The e alua ion me ics consis o ou dimensions: (1) Goodness o i ; (2)
Consis ency o bi a ia e ela ionships; (3) Accu acy in cap u ing co ne solu ions; and (4)
Accu acy in holding cons ain s. They a e calcula ed o di e en sizes o aining se used
o aining o unde s and he e o needed in da a gene a ion. In ou speci ic case, inc eas-
ing he size o aining se om 1000 o 50,000 signi ican ly imp o es he pe o mance o all
ypes o models. Once he amoun o aining da a eaches 100,000, adding mo e da a poin s
o aining does no imp o e he pe o mance o he su oga e models in any ele an way as
de ined by he e alua ion me ics. MLP pe o ms he second bes in gene al, and i s pe o -
mance on o he c i e ia is close o he bes model— BiLSTM. Since i has a s ong ad an age
on in e ence ime, MLP migh be he p ime choice o many cases wi h s ong compu a ional
demands.
Ou esea ch shows NNs a e e icien in app oxima ing de ailed a m- le el models. Thus,
hey can o e upscaling possibili ies o ABMs wi h de ailed a m- le el model ou comes.
Speci ically, he in eg a ed modelling sys em can be used o enable comp ehensi e analyses o
ag i- en i onmen al policies ha a e a ge ed a he indi idual a m le el. I will be wo h ex-
plo ing whe he he sligh de ia ion (like 1%) o he su oga e model a he a m le el can cause
c ucial di e gence a he egional le el, whe e he e ogeneous a ms in e ac wi h each o he in
bo h he sho and long un. Fu he mo e, upda ing and debugging he in eg a ed modelling
sys em could be challenging because h ee di e en models (i.e., a m model, su oga e model
and ABM) ha a e po en ially ope a ed by di e en eams a e in ol ed.
Finally, u u e esea ch may mo e owa ds mo e sys ema ic de elopmen and in eg a ed
applica ion o su oga e models going beyond hei s and- alone me hodological assessmen .
An in e es ing al e na i e a enue in aining su oga e models migh be he use o gene a i e
ad e sa ial ne wo ks (GANs) (Good ellow e al.,2014). They could lea n he c i e ia o mak-
ing he ou comes om he o iginal and su oga e model indis inguishable in a da a- d i en way
o could allow us o de i e mo e na u al s opping c i e ia o da a gene a ion. The apid de el-
opmen o machine lea ning will likely u he imp o e he pe o mance o su oga e models
and make he aining o NNs a mo e s anda d app oach.
ACKNO WLE DGE MENTS
This wo k ecei ed unding om he Eu opean Union's esea ch and inno a ion p og amme
unde g an ag eemen No. 817566 – MIND STEP. I is also pa ially unded by he Ge man
Resea ch Founda ion unde Ge many's Excellence S a egy, EXC- 2070- 390732324- PhenoRob.
Ou hanks a e also due o anonymous e iewe s o hei cons uc i e commen s on an ea lie
d a . Open Access unding enabled and o ganized by P ojek DEAL.
DATA AVAILABILITY STATEMENT
The da a and code used o his pape can be ound in he ollowing Gi hub eposi o y: h ps://
gi hub.com/linme ishan g/Su o ga eNN.
14779552, 2024, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1111/1477-9552.12543 by Coch ane Ge many, Wiley Online Lib a y on [14/02/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
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257
SURROGATE MODELLING USING DEEP LEARNING
ORCID
Linmei Shang h ps://o cid.o g/0000-0002-5044-3219
F anziska Appel h ps://o cid.o g/0000-0002-6049-4511
ENDNOTES
1 See Good ellow e al.(2016) o u he de ails.
2 See he Gi hub eposi o y o he ‘Ke as’ lib a y (Cholle ,2015).
3 In he con ex o OLS (o dina y leas squa es),
R2
will always be be ween 0 and 1. Bu in he machine lea ning con ex ,
when
R2
is calcula ed based on a es se no used in es ima ion, nega i e alues may occu e en i he i c i e ion in
aining is leas squa es.
4 In p ac ice, his h eshold is <0.01 because NNs usually do no p edic a s ic ‘0’ bu a he a e y small numbe like
0.000001.
5 This is he maximum amoun o obse a ions in he o iginal aining se .
6 Because o he poo pe o mance, he e alua ions o ResNe wi h 1000 obse a ions a e no shown in he ollowing
igu es, ei he .
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