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

2026-01-23

AI Summary

This work proposes NewPINNs, a novel framework that addresses the well-known optimization failures of traditional physics-informed neural networks (PINNs) when solving partial differential equations (PDEs), which often stem from ill-conditioned residual loss formulations, sensitivity to loss weighting, and challenges posed by stiffness or strong nonlinearity. NewPINNs uniquely integrates classical numerical solvers—such as finite volume, finite element, and spectral methods—directly into the neural network training process. Instead of relying on explicit PDE residual and boundary condition losses, the method enforces consistency between the neural network predictions and the states evolved by the embedded numerical solver through a pull-push interaction mechanism. This approach effectively circumvents common failure modes of PINNs, significantly enhancing solution stability and accuracy across a range of forward and inverse PDE problems, particularly in stiff and highly nonlinear regimes.

Technology Category

• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Physics-Informed Neural Networks
• Partial Differential Equations
• Numerical Solvers
• Optimization Pathologies
• Stiff Problems

Innovation

• Physics-Informed Neural Networks
• Numerical Solvers
• Solver-Consistency
• Differential Equations
• Neural Network Training