Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

2026-05-30

AI Summary

This work addresses the challenge of poor convergence in physics-informed neural networks (PINNs) when solving complex partial differential equations, often caused by ill-conditioned loss landscapes. The authors propose the FK-PINNs framework, which introduces a data fidelity term derived from the Feynman–Kac representation at a small set of points, effectively acting as an operator-level preconditioner. This approach yields the first non-asymptotic L² error bound for PINNs employing tanh activation and establishes a novel bound on the pseudo-dimension of their derivative networks. By integrating the Feynman–Kac stochastic representation, Monte Carlo sampling, and generalization analysis, FK-PINNs significantly outperform standard PINNs on Poisson, Schrödinger, mean first-passage time, and flux problems—successfully resolving cases where conventional PINNs fail.

Abstract

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic $L^2(Ω)$-error bounds for FK-PINNs with $\tanh$ activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

Problem

• Physics-Informed Neural Networks
• loss landscape
• ill-conditioning
• partial differential equations
• Feynman-Kac representation

Innovation

• Physics-Informed Neural Networks
• Feynman-Kac representation
• operator preconditioning
• non-asymptotic error bounds
• loss landscape conditioning