Cautious Conflict-Free Inverse Gradient Methods

Abstract

Partial differential equations (PDEs) are indispensable tools for modeling complex phenomena in physics and engineering. Physics-Informed Neural Networks (PINNs) have emerged as a powerful alternative to traditional numerical solvers by directly incorporating physical constraints into neural network loss functions. Despite their advantages, PINN training requires multi-objective optimization, which often leads to significant gradient conflicts among boundary, initial, and residual loss terms. The Conflict-Free Inverse Gradient (ConFIG) method addresses these conflicts; however, integrating ConFIG with adaptive optimizers such as Adam introduces a critical vulnerability: the conflict-free direction generated by ConFIG is further processed by the optimizer, and the resulting weight update may no longer be conflict-free with respect to the task gradients. To address this issue, Cautious ConFIG is proposed as a concise three-line extension to the ConFIG algorithm that empirically restores the conflict-freeness guarantee prior to the weight update. The method is evaluated on three PINN benchmarks: the 1D Viscous Burgers' equation, the 1D Nonlinear Schr\"odinger equation, and the 2D Kovasznay flow. Experimental results demonstrate consistent performance improvements over the standard ConFIG baseline. Additionally, the stabilizing effect of cautious masking and its reaction to conflict severity, task number, and optimizer lag are investigated using a simple toy system.