Biophysics-informed deep operator learning for inverse problems with application to electrophysiological source reconstruction

Abstract: Electrophysiological brain signals are typically acquired through indirect and noisy measurements, providing transformed representations of the underlying neural activity. Source reconstruction---the inverse problem of resolving underlying neural signals from these measurements---is essential for mapping brain function but remains challenging because it is ill-posed and sensitive to noise. Deep learning methods have shown promise across a range of inverse problems, yet many do not explicitly incorporate the biophysical principles governing data generation, limiting data efficiency and adaptation across subjects. Here, we introduce DeepOp-Informed, a biophysics-informed geometric deep operator learning framework that embeds the biophysics of the sensing process into the model through a custom differentiable layer, enabling more efficient learning and improved reconstruction performance. This layer enables the neural network to adapt to subject-specific variations in the physics of signal generation, resulting from differences in brain anatomy and sensor positioning. In realistic magnetoencephalography simulations, DeepOp-Informed generalizes to forward models from held-out subjects, reducing reconstruction error relative to several neural-network and classical baselines. Applied to adolescent auditory-evoked recordings, it produces anatomically plausible reconstructions localized to the auditory cortex. While our application focuses on magnetoencephalography, the framework is general and may be adaptable to other imaging modalities.

• The paper introduces DeepOp-Informed, a differentiable forward-model-conditioned layer combined with a cortical Graph U-Net to solve ill-posed MEG source reconstruction while directly incorporating subject-specific biophysics.
• The method reduced normalized reconstruction error by approximately threefold versus nine classical and neural-network baselines and matched or exceeded competitors using only 200 synthetic training samples.
• The framework generalized across held-out subjects and produced physiologically plausible auditory-cortex localization in real adolescent MEG data, although sensitivity to forward-model errors and the lack of ground-truth validation remain limitations.

DeepOp-Informed is a biophysics-informed geometric deep operator learning framework for solving ill-posed inverse problems, developed and evaluated in the context of magnetoencephalography (MEG) source reconstruction (2608.16871). The central idea is to embed the physics of the sensing process directly into a neural network architecture through a differentiable custom layer that conditions on each sample's forward model, rather than requiring the network to learn the source-sensor relationship implicitly from data pairs. This design yields substantial gains in data efficiency and cross-subject generalization: on realistic MEG simulations, the method achieves roughly a threefold reduction in reconstruction error relative to nine neural-network and classical baselines while training on only 200 synthetic samples.

Motivation and problem setting

MEG source reconstruction maps extracranial magnetic measurements to the spatial distribution of neuronal currents on the cortex. Under the quasi-static approximation to Maxwell's equations, the measurement model is linear: $y_i = \mathcal{K}_i x_i + \epsilon_i$, where $x_i \in \mathbb{R}^p$ is the source signal over $p$ cortical dipoles, $y_i \in \mathbb{R}^s$ is the sensor signal, $\mathcal{K}_i \in \mathbb{R}^{s \times p}$ is a subject-specific forward model encoding head anatomy and sensor geometry, and $\epsilon_i$ is noise. Because $s \ll p$ and the forward operator mixes sources non-locally, the inverse problem is severely ill-posed. Classical regularized least-squares methods (MNE, dSPM, sLORETA, eLORETA, LCMV) are linear in the data and therefore limited in expressiveness; existing neural-network approaches to the M/EEG inverse problem learn the forward physics only implicitly from synthetic source-sensor pairs, which is data-inefficient and can produce unstable reconstructions, consistent with known instabilities of learned inverse solvers.

The key architectural distinction of DeepOp-Informed is that it treats the forward model as an input to the network rather than as something to be inferred. Once trained, the same parameters can be deployed across subjects by supplying each subject's own $\mathcal{K}$, accommodating differences in brain geometry, sensor positioning, and even the number of usable channels.

Architecture

The reconstruction operator $$\mathcal{R}_{\theta,\tilde{\mathcal K}} = \mathcal{F}_{\theta_2} \circ \mathcal{B}_{\theta_1,\tilde{\mathcal K}}$$ comprises two modules trained end-to-end with an MSE loss on synthetic triplets $(y_i, x_i, \tilde{\mathcal K}_i)$.

Biophysics-informed layer. The layer solves, per sample,

$x_i \in \mathbb{R}^p$0

where the sparse penalty matrix is parameterized as $x_i \in \mathbb{R}^p$1, with $x_i \in \mathbb{R}^p$2 the finite-element discretization of the Laplace–Beltrami operator on the cortical mesh. Under a working joint Gaussian assumption, this output equals the conditional expectation $x_i \in \mathbb{R}^p$3, so the layer admits an empirical Bayes interpretation with a learned prior precision. The authors position this as a generalization of a convolutional layer: linear, low-dimensional, differentiable, but defining a global kernel that disentangles the mixing introduced by the forward model — unlike a fixed pseudoinverse back-projection, whose suboptimality their simulations confirm.

Refinement module. A Graph U-Net operating on the cortical mesh graph refines the preliminary estimate using anatomical structure. A multi-head extension stacks several biophysics-informed layers in parallel with distinct learned penalties, concatenating outputs as node features; this improves performance particularly at higher SNR, though results are mixed at lower SNR, so the number of heads must be treated as a tuning parameter.

A notable computational contribution is that both forward and backward passes reduce to sparse linear systems. Proposition 1 shows the layer output is obtained from a saddle-point system with block matrix involving $x_i \in \mathbb{R}^p$4, $x_i \in \mathbb{R}^p$5, and $x_i \in \mathbb{R}^p$6; Proposition 2 shows the gradient with respect to $x_i \in \mathbb{R}^p$7 reduces to an outer product once an auxiliary system (structurally identical to the forward pass) is solved. Both are implemented as a custom differentiable PyTorch operation.

Simulation results

The realistic simulation setting used forward models from 19 adolescents (204 gradiometers, $x_i \in \mathbb{R}^p$8 cortical vertices), with training on 13 subjects and testing on 6 held-out subjects, realistic noise covariances estimated from prestimulus recordings, and SNRs averaging 8.5 and 4.8. Across 20 Monte Carlo repetitions and varying numbers of coactive sources, DeepOp-Informed achieved approximately a threefold reduction in normalized reconstruction error relative to all baselines: MLP, GraphU (fully learned), GraphUBP (back-projection plus Graph U-Net), an unrolled MODL-style architecture, and five classical linear methods. The advantage also held under AUC-based evaluation of active-region detection via paired $x_i \in \mathbb{R}^p$9-tests, indicating the gain is not merely an artifact of support recovery.

Several secondary findings sharpen the interpretation. Providing an incorrect forward model visibly changes the DeepOp-Informed reconstruction, confirming the model actively exploits subject-specific physics rather than ignoring its conditioning input. Physics-agnostic models (MLP, GraphU) performed worst in the multi-subject setting but improved when train and test shared a fixed forward model, isolating cross-subject generalization as the specific failure mode the informed layer addresses. The unrolled architecture, designed for higher-SNR MRI settings, performed poorly here, although the authors concede that alternative adaptations might fare better. In controlled spherical-source experiments, the data-efficiency gap was stark: MLP did not match DeepOp-Informed even with 20,000 training samples, and GraphU required more than 5,000–10,000 samples depending on SNR to match DeepOp-Informed's performance with 200. Robustness experiments showed graceful degradation under forward-model perturbation up to moderate magnitudes, with faster deterioration at higher SNR — a limitation worth noting, since high-SNR regimes amplify sensitivity to forward-model error.

Application to adolescent MEG recordings

Applied to auditory-evoked responses (M100 component, averaged over 420 trials of trisyllabic pseudoword streams), models trained purely on synthetic data produced physiologically plausible reconstructions. In the multi-subject setting with a held-out subject, DeepOp-Informed yielded activation sharply localized to the auditory cortex, whereas MLP, GraphU, and the unrolled network deviated substantially from expected patterns, and classical methods identified the auditory cortex but with activation spread over broader areas more reflective of proximity to high-amplitude sensors than of functional organization. Reconstructions of the second and third syllable responses — perturbations of the first — were stable, supporting robustness. Single-subject-trained models produced even sharper localization, plausibly because they avoid registration error and forward-model uncertainty, but require per-subject retraining, which limits clinical scalability. It should be emphasized that no ground truth exists in this real-data evaluation; the assessment rests on physiological plausibility rather than quantitative accuracy.

Limitations and open questions

The paper is candid about several constraints. The Gaussian working assumption underlying the empirical Bayes interpretation of the biophysics-informed layer is approximate, and the true source covariance structure is learned only through the low-dimensional Laplacian-power parameterization. Robustness to forward-model misspecification degrades at high SNR, and head-movement artifacts are not modeled; the authors propose incorporating movement into synthetic training data and comparing against signal-space separation preprocessing as open work. Uncertainty quantification for reconstructions, explicit use of subject-specific cortical geometry in the multi-subject model, and validation on phantom data remain unaddressed. Whether the framework transfers to other imaging modalities — MRI, CT, PET — or to tasks beyond reconstruction, such as classification, is asserted as feasible but not demonstrated.

Conclusion

DeepOp-Informed demonstrates that embedding a differentiable, forward-model-conditioned regularized inversion layer into a geometric deep learning pipeline materially improves data efficiency, cross-subject generalization, and reconstruction fidelity for electrophysiological source imaging, with a threefold error reduction over strong baselines in realistic simulations and anatomically plausible localization in real MEG recordings from as few as 200 training samples. The framework's generality — any task-specific refinement module can replace the Graph U-Net — makes it a candidate template for physics-informed operator learning in other ill-posed imaging problems, though its performance boundaries under severe forward-model error and in modalities with different noise structure remain to be established.