PhiDelayNe : A Physics-In o med Neu al Ne wo k
o Time-Se ies wi h Va iable Delays
Bahman Masa a
Independen Resea che
Email: [email p o ec ed]
Abs ac —This pape in oduces PhiDelayNe , a physics-
in o med neu al ne wo k designed o o ecas ime se ies wi h
nonlinea , a iable delays. The co e o he model is a dynamic
delay ope a o , τ(x, ), inspi ed by 5D scala ield heo y, whose
componen s a e alida ed h ough an abla ion s udy. The model’s
e icacy was demons a ed h ough a wo-s age p ocess. Fi s , on
a syn he ic benchma k, i achie ed a 37.5% educ ion in Mean
Squa ed E o o e a s anda d LSTM. Second, when applied o a
eal-wo ld hyd ology da ase , he model’s in e p e able ope a o
success ully unco e ed a non-ob ious, physically meaning ul
insigh : he sys em’s esponse delay is highe du ing complex
we seasons han du ing simple d y seasons. This alida es he
model’s abili y o no only p edic accu a ely bu also o p o ide
ac ionable insigh s o eal-wo ld applica ions like wa e esou ce
managemen . The code o his p ojec is publicly a ailable on
Gi Hub.
Index Te ms—physics-in o med neu al ne wo ks, ime-se ies
o ecas ing, a iable delay, in e p e able AI, abla ion s udy,
hyd ology.
I. INTRODUCTION
Time-se ies o ecas ing is a undamen al ask in many
scien i ic and indus ial domains. While deep lea ning models
like Recu en Neu al Ne wo ks (RNNs) ha e achie ed s a e-
o - he-a pe o mance, hey o en s uggle wi h a speci ic
challenge: **nonlinea , a iable ime delays**.
To add ess his gap, we p opose PhiDelayNe 1, a no el a -
chi ec u e ha in eg a es a dynamic, in e p e able, and physics-
in o med delay ope a o di ec ly in o he ne wo k. This pape
p esen s i s heo e ical ounda ion, implemen a ion, and a
igo ous alida ion on bo h syn he ic and eal-wo ld da a. We
hen discuss he p ac ical implica ions o ou indings and
conclude wi h he model’s o e all con ibu ion.
II. THEORETICAL FRAMEWORK
The cen al inno a ion o PhiDelayNe is he dynamic delay
ope a o τi(x, ), o mula ed as an e ec i e scala ield ˜
ϕ(x, ).
The o mula ion is:
τi(x, ) = exp (−Va [x( −i: )])·"˜
ϕ0+
N
X
n=1
Ane−ni/R
i+ϕγ#
(1)
whe e he exponen ial e m p o ides a iance-based adap-
a ion, ˜
ϕ0is a lea nable ze o-mode e m, he summa ion
ep esen s highe -mode Yukawa-like con ibu ions inspi ed by
5D physics [1], and ϕγaccoun s o pho onic e ec s [2].
1Code and da a a ailable a : h ps://gi hub.com/bahman2017/PhiDelayNe
III. METHODOLOGY
The PhiDelayNe model was implemen ed in PyTo ch. The
co e a chi ec u e consis s o a ‘DynamicDelayModule‘ ha
compu es τacco ding o Eq. 1. The inpu sequence is hen
weigh ed by he compu ed delay and ed in o a 2-laye LSTM.
The model is ained using a cus om physics-in o med loss
unc ion ha combines Mean Squa ed E o (MSE) wi h a
smoo hness cons ain on he lea ned delay e m.
IV. EXPERIMENTAL VALIDATION
We conduc ed a se ies o expe imen s o alida e PhiDe-
layNe ’s pe o mance, in e p e abili y, and a chi ec u al de-
sign.
A. Syn he ic Da a Expe imen
We i s es ed he model on a syn he ic da ase wi h a
known, oscilla ing g ound- u h delay. PhiDelayNe decisi ely
ou pe o med he LSTM baseline, educing he MSE by 37.5%
(Table I).
TABLE I
PERFORMANCE ON THE SYNTHETIC TEST SET
Model MSE MAE R²
LSTM Baseline 0.2137 0.3775 0.5716
PhiDelayNe 0.1335 0.3065 0.7323
B. Real-Wo ld Hyd ology Expe imen
We hen applied he model o p edic daily i e discha ge
om p ecipi a ion da a o he Ame ican Ri e , CA (2010-
2020). The model p oduced empo ally accu a e o ecas s
and, mo e impo an ly, a physically meaning ul lea ned delay
pa ame e τ(Figu e 1).
C. Abla ion S udy
To jus i y he model’s complexi y, we conduc ed an abla ion
s udy. As shown in Figu e 2, emo ing he physics-inspi ed
componen s (Yukawa Modes and Pho onic Te m) signi ican ly
deg aded pe o mance, con i ming hei c i ical ole and ali-
da ing he a chi ec u al design.
Fig. 1. Resul s on he eal-wo ld hyd ology da ase . (Top) PhiDelayNe ’s p edic ions (g een) show supe io empo al accu acy on discha ge peaks compa ed
o he LSTM baseline (pu ple). (Bo om) The model’s lea ned delay pa ame e τexhibi s clea seasonal pa e ns, demons a ing he model’s in e p e abili y.
Fig. 2. Abla ion s udy esul s. The plo shows he inc ease in Tes MSE as
componen s a e emo ed om he ull PhiDelayNe model.
V. DISCUSSION: IMPLICATIONS FOR WATER RESOURCE
MANAGEMENT
The success ul applica ion o PhiDelayNe o he hyd ology
da ase o e s mo e han jus an accu a e o ecas ; i p o ides
a new, in e p e able ool wi h signi ican implica ions o
p ac ical wa e esou ce managemen .
The model’s lea ned delay pa ame e , τ, e ec i ely ac s as
a dynamic, eal- ime indica o o he wa e shed’s o e all s a e.
Ou analysis showed ha τis no s a ic bu a ies seasonally,
e ealing a physically meaning ul insigh : he a e age lea ned
delay is highe du ing he we season han he d y season. This
coun e -in ui i e esul e lec s he complex physical p ocesses
o he wa e shed.
This in e p e abili y ansla es in o wo key bene i s:
1) Enhanced Ope a ional Fo ecas ing: The model’s su-
pe io empo al accu acy allows o mo e eliable lood
wa nings and op imized dam ope a ions.
2) A New Diagnos ic Tool o Wa e shed Heal h: The
lea ned delay τcan be moni o ed o e he long e m o
de ec changes in he en i onmen due o u baniza ion
o wild i es.
VI. CODE AVAILABILITY
The sou ce code o he PhiDelayNe model, along wi h
he sc ip s o ep oduce he expe imen s in his pape , is pub-
licly a ailable on Gi Hub a : h ps://gi hub.com/bahman2017/
PhiDelayNe .
VII. CONCLUSION
This pape in oduced PhiDelayNe , a no el, physics-
in o med neu al ne wo k designed o o ecas ime se ies wi h
a iable delays. We ha e shown h ough syn he ic da a, a
eal-wo ld applica ion, and a de ailed abla ion s udy ha i s
a chi ec u e is e ec i e, in e p e able, and well-jus i ied. I s
abili y o p o ide physically meaning ul insigh s highligh s
i s po en ial as a ool no jus o p edic ion, bu o scien-
i ic disco e y and p ac ical decision suppo . By embedding
physical p inciples di ec ly in o i s a chi ec u e, PhiDelayNe
o e s a powe ul and obus al e na i e o black-box models,
pa ing he way o mo e in elligen and explainable ime-se ies
analysis.
REFERENCES
[1] P. Bull, e al., ”Beyond ΛCDM: P oblems, solu ions, and he oad ahead,”
Physics o he Da k Uni e se, ol. 12, pp. 56–99, 2016.
[2] B. Masa a , ”In eg a ing a 5D Scala Field wi h Gene al Rela i i y and
Pho onic Con ibu ions,” P ep in , Zenodo, 2025. [Online]. A ailable:
h ps://zenodo.o g/ eco ds/16667924
