🤖 AI Summary
Physics-informed neural networks (PINNs) minimize the mean residual, which can obscure localized large errors and compromise solution reliability. To address this, we propose a residual risk-aware training framework that explicitly targets tail residual control. Our method introduces conditional value-at-risk (CVaR) and mean excess (ME) surrogate penalties, establishing a theoretical connection between risk-sensitive optimization and chance-constrained learning. We further design three mechanisms: persistent hard-point replay, local risk budgeting, and multi-objective risk modeling—enabling explicit trade-offs between accuracy and tail robustness. Experiments on Burgers, heat, Korteweg–de Vries (KdV), and Poisson equations demonstrate that our approach reduces tail errors by up to 57%, maintains or improves mean error, enhances training stability, and effectively handles discontinuous solutions.
📝 Abstract
Physics-informed neural networks (PINNs) typically minimize average residuals, which can conceal large, localized errors. We propose Residual Risk-Aware Physics-Informed Neural Networks PINNs (RRaPINNs), a single-network framework that optimizes tail-focused objectives using Conditional Value-at-Risk (CVaR), we also introduced a Mean-Excess (ME) surrogate penalty to directly control worst-case PDE residuals. This casts PINN training as risk-sensitive optimization and links it to chance-constrained formulations. The method is effective and simple to implement. Across several partial differential equations (PDEs) such as Burgers, Heat, Korteweg-de-Vries, and Poisson (including a Poisson interface problem with a source jump at x=0.5) equations, RRaPINNs reduce tail residuals while maintaining or improving mean errors compared to vanilla PINNs, Residual-Based Attention and its variant using convolution weighting; the ME surrogate yields smoother optimization than a direct CVaR hinge. The chance constraint reliability level $α$ acts as a transparent knob trading bulk accuracy (lower $α$ ) for stricter tail control (higher $α$ ). We discuss the framework limitations, including memoryless sampling, global-only tail budgeting, and residual-centric risk, and outline remedies via persistent hard-point replay, local risk budgets, and multi-objective risk over BC/IC terms. RRaPINNs offer a practical path to reliability-aware scientific ML for both smooth and discontinuous PDEs.