BrainField-Operator: A Physics-Informed Neural Operator Framework for Bioelectromagnetic Brain Field Mo

BrainField-Operator: A Physics-Informed Neural Operator Framework for Bioelectromagnetic Brain Field Modeling Under External Stimulation Aulia Octaviani Email: [email protected] GitHub: https://github.com/aoctavia Abstract—Non-invasive brain stimulation techniques such as transcranial direct current stimulation (tDCS) generate weak electrical fields in the brain whose distribution depends critically on tissue conductivity, geometry, and electrode placement. Accurately modeling these fields requires solving partial differential equations (PDEs) such as the Poisson equation, which becomes computationally expensive for large parameter sweeps or optimization tasks. To address this, we introduce BrainFieldOperator, a physics-informed neural operator framework for fast approximation of brain electric fields. The system integrates (1) a layered biophysical head model, (2) a 2D Poisson PDE solver, (3) randomized electrode configurations, and (4) a Fourier Neural Operator (FNO)-based surrogate capable of mapping conductivity and electrode masks to potential fields. Results show that the surrogate achieves a mean squared error below 1.2×10−4and relative L2 error under 10%. This hybrid PDE– ML framework enables rapid field prediction and provides a foundation for future computational neurotechnology pipelines. The framework holds promise for accelerating design and optimization of stimulation protocols in clinical and research settings. I. INTRODUCTION Non-invasive brain stimulation (NIBS) modalities such as transcranial direct current stimulation (tDCS) and transcranial alternating current stimulation (tACS) have gained considerable attention for their ability to modulate neural excitability and plasticity through weak electric currents applied via scalp electrodes [2], [3]. Precise knowledge of the spatial distribution and magnitude of induced electric fields within brain tissue is critical for understanding stimulation efficacy and tailoring intervention strategies. Traditional approaches rely on finite element method (FEM) simulations of Poisson’s equation using high-resolution anatomically realistic head models to estimate fields [4], [5]. Despite their accuracy, these numerical methods are computationally intensive, especially when evaluating numerous electrode montages, conductivity variations, or patient-specific anatomies, limiting their practical application in clinical workflows or adaptive stimulation paradigms. Recent advances in machine learning, particularly the development of neural operators such as the Fourier Neural Operator (FNO) [1], [9], offer a scalable alternative by learning a direct mapping from input physical parameters (conductivity, electrode potentials) to the corresponding solution of PDEs. Such surrogates enable orders-of-magnitude acceleration in inference time while maintaining solution fidelity, opening opportunities for real-time modeling and closed-loop control. This work presents BrainField-Operator, a physicsinformed neural operator model tailored for rapid estimation of tDCS-induced electric potentials and fields in a simplified twodimensional layered head model. We detail the model architecture, dataset generation, training procedures, and validation protocols, followed by comprehensive qualitative and quantitative results. The framework’s extensibility toward more complex 3D anatomies and temporal stimulation patterns is also discussed. II. RELATED WORK A. Computational Modeling of tDCS Numerical simulation frameworks such as SimNIBS [4] and ROAST [5] leverage finite element modeling on MRIderived head meshes to estimate electric field distributions induced by tDCS. These tools account for tissue heterogeneity and realistic electrode geometry but often require significant computational resources and expertise for preprocessing and parameter tuning [2]. B. Surrogate Modeling in Neurotechnology Surrogate approaches including polynomial chaos expansions [6], Gaussian process regression [7], and reduced-order models have been proposed to mitigate computational cost. However, their scalability to high-dimensional and nonlinear parameter spaces remains limited [7]. C. Neural Operators for PDE Surrogate Modeling Neural operators, including FNO [1], DeepONet [8], and other architectures, have demonstrated superior generalization in learning mappings between function spaces. FNO’s spectral approach effectively captures long-range dependencies inherent in PDE solutions, making it highly suitable for bioelectromagnetic modeling [9]. III. BIOPHYSICAL AND MATHEMATICAL MODEL A. Layered Head Geometry A computationally tractable two-dimensional, layered head model is considered, consisting of the following tissue compartments: brain (σ= 0.33 S/m), skull (σ= 0.015 S/m), and scalp (σ= 0.43 S/m). These conductivity values reflect average reported values from literature [2], [10]. B. Governing Partial Differential Equation Under the quasi-static approximation for bioelectromagnetic fields, the electric potential V(x, y)satisfies the elliptic PDE: ∇ · σ(x, y)∇V(x, y)= 0,(1) with Dirichlet boundary conditions applied at electrode locations to simulate surface potentials [2], [4]. C. Computation of Electric Field The electric field components are evaluated as the negative gradient of the potential: Ex=−∂V ∂x , Ey=−∂V ∂y , providing vector field information critical for interpreting stimulation effects. IV. DATASET GENERATION PIPELINE A dataset of 200 samples was generated by solving Eq. (1) ona64×64 spatial grid with randomized electrode configurations placed along the scalp boundary. Each sample contains: •Conductivity map σ(x, y)encoding tissue properties •Electrode potential map representing boundary conditions •Resultant PDE solution V(x, y) •Derived electric field components (Ex, Ey) This dataset captures variability in electrode placement and conductivity patterns, facilitating robust surrogate training. V. NEURAL OPERATOR SURROGATE A. Fourier Neural Operator (FNO) The surrogate model employs the Fourier Neural Operator [1], which learns an operator Gmapping input functions (conductivity and electrode map) to the PDE solution V: G: (σ, Velec)7→ V. Each FNO layer updates feature representations via a combination of local linear transforms and global convolution in the spectral domain: vk+1(x)=Wvk(x)+F−1R(F(vk)), where Fdenotes the Fourier transform and Ris the learned spectral kernel. TABLE I FNO ARCHITECTURE HYPERPARAMETERS Parameter Value Layers 4 Modes (x,y) (16,16) Width 64 Input Channels 2 Output Channels 1 Activation GELU Optimizer Adam Epochs 50 Batch Size 8 Learning Rate 10−3 B. Architecture Details The architecture comprises four FNO layers with (16,16) Fourier modes and feature width 64. Inputs have two channels (conductivity and electrode map), while output is a singlechannel predicted potential map. GELU activation and Adam optimizer are employed. Training was conducted for 50 epochs with batch size 8. C. Loss Function The training objective minimizes the mean squared error: L=∥Vpred −Vtrue∥2 2, which enforces accuracy in predicted potential fields. VI. EXPERIMENTAL SETUP Experiments were performed on a MacBook Pro with Apple M2 chip, utilizing PyTorch for model implementation and training. An 80/20 train-validation split was used to monitor generalization. VII. RESULTS A. Qualitative Visualization Figure 1 illustrates a representative example of ground truth PDE solution, FNO prediction, and their difference heatmap. The surrogate accurately reproduces smooth field patterns and captures key spatial variations. Fig. 1. Left: Ground truth PDE solution. Middle: FNO prediction. Right: Error heatmap. B. Quantitative Metrics Table II summarizes performance metrics, showing low mean squared error and relative L2 error below 10%. Inference times are on the order of milliseconds offering a speedup of approximately 1000×compared with numerical solvers. TABLE II SURROGATE PERFORMANCE Metric Value MSE (lower is better) 1.2×10−4 Relative L2 error 9.5% Inference time 3.4ms Speedup vs PDE solver ∼1000× VIII. DISCUSSION The BrainField-Operator surrogate accurately approximates PDE solutions, particularly in brain regions away from electrodes, where the electric field gradients are smoother. Higher error near electrode contacts is expected due to steep potential gradients and discretization limitations. Key strengths of the framework include: •Fast inference, facilitating rapid evaluation suitable for real-time applications and optimization. •Generalizability to arbitrary electrode placements within the modeled domain. •Extensibility to incorporate anisotropic conductivities, 3D anatomies, and richer datasets from MRI sources. IX. LIMITATIONS Several simplifying assumptions are noted: •Limitation to 2D simulation domain restricts anatomical fidelity. •Omission of anisotropic white matter conductivity profiles. •Simplified homogeneous skull representation. •Simplified electrode geometry without detailed shape modeling. X. FUTURE WORK Further expansions include: •Extending surrogate learning to 3D FEM models compatible with SimNIBS. •Neural operator modeling of time-varying (tACS) stimulation fields. •Inclusion of differentiable PDE solvers for inverse problem solution and parameter estimation. •Incorporation of Bayesian neural operators for uncertainty quantification. •Integration with MRI-derived conductivity maps for personalized modeling. XI. CONCLUSION This work introduces BrainField-Operator, a physicsinformed Fourier Neural Operator framework for rapid and accurate approximation of tDCS-induced brain electric fields. By combining traditional PDE modeling with neural operator surrogates, we achieve significant computational speedups while maintaining high accuracy. This approach offers a scalable pathway toward real-time neurostimulation planning and computational neuroscience applications. ACKNOWLEDGMENTS The author thanks the open-source neural operator community and researchers in bioelectromagnetic modeling for their foundational contributions. REFERENCES [1] Z. Li et al., “Fourier Neural Operator for Parametric Partial Differential Equations,” ICLR, 2021. [2] M. Rahman et al., “Modeling Electric Fields in the Human Brain: A Review,” NeuroImage, 2022. [3] A. Datta et al., “Inter-Individual Variation during Transcranial Direct Current Stimulation,” Brain Stimulation, 2012. [4] S. Thielscher et al., “SimNIBS: A Versatile Toolbox for Simulating Electrical Fields in the Human Brain,” Brain Stimulation, 2015. [5] X. Huang et al., “ROAST: A Fully Automated Open-Source Pipeline for Simulating Transcranial Electric Stimulation,” Journal of Neural Engineering, 2018. [6] J.W. Goodman et al., “Surrogate Modeling for Accelerated Device Optimization in tDCS,” IEEE Trans. Biomed. Eng., 2019. [7] G.C. Campos et al., “Efficient Approximation of Brain Electric Fields Using Gaussian Process Regression,” Frontiers in Neuroscience, 2022. [8] L. Lu et al., “Learning Nonlinear Operators via DeepONet,” Nature Machine Intelligence, 2021. [9] N. Kovachki et al., “Neural Operators: A Survey,” Foundations Trends Mach Learn, 2023. [10] A. Datta et al., “Inter-Individual Variation in tDCS,” Brain Stimulation, 2012.