NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations

TL;DR

NewPINNs is a novel framework that integrates traditional numerical solvers into neural network training for solving PDEs, improving stability and accuracy over standard physics-informed neural networks.

Contribution

It introduces a solver-coupled training approach that reduces the need for problem-specific loss engineering and enhances performance in complex PDE regimes.

Findings

Mitigates common PINN failure modes

Effective across multiple solver types

Improves stability in stiff and nonlinear problems

Abstract

We introduce NewPINNs, a physics-informing learning framework that couples neural networks with conventional numerical solvers for solving differential equations. Rather than enforcing governing equations and boundary conditions through residual-based loss terms, NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver-consistency. The neural network produces candidate solution states that are advanced by the numerical solver, and training minimizes the discrepancy between the network prediction and the solver-evolved state. This pull-push interaction enables the network to learn physically admissible solutions through repeated exposure to the solver's action, without requiring problem-specific loss engineering or explicit evaluation of differential equation residuals. By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes. We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.