On the Apparent Pareto Front of Physics-informed Neural Networks

Physics-informed neural networks (PINNs) have emerged as a promising deep learning method, capable of solving forward and inverse problems governed by differential equations. Despite their recent advance, it is widely acknowledged that PINNs are difficult to train often require careful tuning loss weights when data physics functions combined scalarization multi-objective (MO) problem. In this paper, we aim understand how parameters the physical system, such characteristic length time scales, computational domain, coefficients equations affect MO optimization optimal choice weights. Through theoretical examination where these system appear in PINN training, find they effectively individually scale residuals, causing imbalances with certain choices parameters. The immediate effects reflected apparent Pareto front, which define set values achievable gradient-based training visualize accordingly. We empirically verify can be used successfully compensate for scaling parameters, enable selection an solution on front aligns well physically valid solution. further demonstrate altering parameterization, shift exhibit locally convex parts, resulting wider range becomes successful. This work explains PINNs, highlights utility proposed weighting schemes.