A Generalizable Physics-informed Learning Framework for Risk Probability Estimation

TL;DR

This paper introduces a physics-informed neural network framework that efficiently estimates long-term risk probabilities and their gradients, outperforming traditional Monte Carlo methods in accuracy, generalization, and adaptability for safe control applications.

Contribution

It develops a novel PDE-based approach combining MC methods and neural networks to improve risk probability estimation and gradient computation in dynamic environments.

Findings

Better sample efficiency than Monte Carlo methods

Generalizes well to unseen regions

Accurately estimates risk probability gradients

Abstract

Accurate estimates of long-term risk probabilities and their gradients are critical for many stochastic safe control methods. However, computing such risk probabilities in real-time and in unseen or changing environments is challenging. Monte Carlo (MC) methods cannot accurately evaluate the probabilities and their gradients as an infinitesimal devisor can amplify the sampling noise. In this paper, we develop an efficient method to evaluate the probabilities of long-term risk and their gradients. The proposed method exploits the fact that long-term risk probability satisfies certain partial differential equations (PDEs), which characterize the neighboring relations between the probabilities, to integrate MC methods and physics-informed neural networks. We provide theoretical guarantees of the estimation error given certain choices of training configurations. Numerical results show the proposed method has better sample efficiency, generalizes well to unseen regions, and can adapt to systems with changing parameters. The proposed method can also accurately estimate the gradients of risk probabilities, which enables first- and second-order techniques on risk probabilities to be used for learning and control.