# RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2511.18515 · HTML: https://mlsanity.com/paper/2511.18515/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 23 Nov 2025.

## Builds on (most cited first, 13 of 13)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Learning with Average Top-k Loss](https://mlsanity.com/paper/1705.08826/) (2017)
- [Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective](https://mlsanity.com/paper/2506.19805/) (2025)

## Abstract (arXiv)

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.