Title: Mitigating Loss Landscape Difficulties in PINNs via Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

Physics-Informed Neural Networks (PINNs) frequently struggle with slow convergence or outright failure when applied to difficult partial differential equations (PDEs). Recent research has attributed these issues to highly ill-conditioned loss landscapes, which stem directly from the properties of the underlying differential operators. In this work, we investigate PINNs enhanced by a pointwise data-fidelity term, which is incorporated at a limited number of locations within the domain alongside the conventional residual and boundary losses. We demonstrate that this additional supervision functions as an operator-level preconditioner. Specifically, when appropriate weights are selected, our comparative analysis ensures that the condition number is significantly reduced compared to standard PINN formulations, a benefit that holds regardless of the method used to generate the pointwise labels.

For a wide range of PDEs that possess a Feynman-Kac (FK) representation, we produce these labels using Monte Carlo averages of the FK functional, a method we term "FK-PINNs." By employing excess risk decomposition, we derive non-asymptotic $L^2(\Omega)$-error bounds for FK-PINNs utilizing $\tanh$ activation functions and trained via a finite number of gradient descent steps. During this analysis, we also establish pseudo-dimension bounds for the first- and second-order derivatives of $\tanh$ neural networks. To the best of our knowledge, these bounds are novel and hold independent significance. Numerical evaluations on Poisson, Schrödinger, mean exit time, and committor problems validate our theoretical findings, demonstrating that FK-PINNs can effectively resolve PDEs where standard PINNs typically encounter severe failure modes.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC