RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks

TL;DR

RRaPINNs introduce a risk-sensitive framework for physics-informed neural networks that effectively reduces large, localized residual errors by focusing on tail residuals using CVaR and a novel ME surrogate, improving reliability in solving PDEs.

Contribution

The paper proposes RRaPINNs, a novel risk-aware PINN framework that optimizes tail residuals with CVaR and ME surrogate, enhancing the control of worst-case PDE residuals.

Findings

RRaPINNs reduce tail residuals across various PDEs.

The ME surrogate provides smoother optimization than CVaR hinge.

Risk level $oldsymbol{eta}$ effectively balances accuracy and tail control.

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 $\alpha$ acts as a transparent knob trading bulk accuracy (lower $\alpha$ ) for stricter tail control (higher $\alpha$ ). 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.