Overcoming the Loss Conditioning Bottleneck in Optimization-Based PDE Solvers: A Novel Well-Conditioned Loss Function

Abstract

Optimization-based PDE solvers that minimize scalar loss functions have gained increasing attention in recent years. These methods either define the loss directly over discrete variables, as in Optimizing a Discrete Loss (ODIL), or indirectly through a neural network surrogate, as in Physics-Informed Neural Networks (PINNs). However, despite their promise, such methods often converge much more slowly than classical iterative solvers and are commonly regarded as inefficient. This work provides a theoretical insight, attributing the inefficiency to the use of the mean squared error (MSE) loss, which implicitly forms the normal equations, squares the condition number, and severely impairs optimization. To address this, we propose a novel Stabilized Gradient Residual (SGR) loss. By tuning a weight parameter, it flexibly modulates the condition number between the original system and its normal equations, while reducing to the MSE loss in the limiting case. We systematically benchmark the convergence behavior and optimization stability of the SGR loss within both the ODIL framework and PINNs-employing either numerical or automatic differentiation-and compare its performance against classical iterative solvers. Numerical experiments on a range of benchmark problems demonstrate that, within the ODIL framework, the proposed SGR loss achieves orders-of-magnitude faster convergence than the MSE loss. Further validation within the PINNs framework shows that, despite the high nonlinearity of neural networks, SGR consistently outperforms the MSE loss. These theoretical and empirical findings help bridge the performance gap between classical iterative solvers and optimization-based solvers, highlighting the central role of loss conditioning, and provide key insights for the design of more efficient PDE solvers.