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Physics-Informed Representation and Learning: Control and Risk Quantification

Zhuoyuan Wang, Reece Keller, Xiyu Deng, Kenta Hoshino, Takashi Tanaka, Yorie Nakahira

TL;DR

This work tackles efficient stochastic optimal and safety-critical control for high-dimensional systems by combining a comparison-theorem–driven dimensionality reduction with physics-informed neural networks to solve reduced PDEs for the value function and long-horizon safety probability. The key idea is to identify low-dimensional feature coordinates that exactly capture the evolution of the cost and barrier quantities, transform the problem into low-dimensional PDEs via the Feynman-Kac framework, and solve these PDEs with PINNs that integrate data and physics. A dedicated autoencoder-like network automatically discovers meaningful features that satisfy the required bounding assumptions, enabling scalable generalization to unseen regions and longer horizons. Empirical results on synthetic high-dimensional systems show substantially better sample efficiency than path-integral Monte Carlo and successful feature learning, underscoring the approach’s potential for real-world high-dimensional control and risk quantification tasks.

Abstract

Optimal and safety-critical control are fundamental problems for stochastic systems, and are widely considered in real-world scenarios such as robotic manipulation and autonomous driving. In this paper, we consider the problem of efficiently finding optimal and safe control for high-dimensional systems. Specifically, we propose to use dimensionality reduction techniques from a comparison theorem for stochastic differential equations together with a generalizable physics-informed neural network to estimate the optimal value function and the safety probability of the system. The proposed framework results in substantial sample efficiency improvement compared to existing methods. We further develop an autoencoder-like neural network to automatically identify the low-dimensional features of the system to enhance the ease of design for system integration. We also provide experiments and quantitative analysis to validate the efficacy of the proposed method. Source code is available at https://github.com/jacobwang925/path-integral-PINN.

Physics-Informed Representation and Learning: Control and Risk Quantification

TL;DR

This work tackles efficient stochastic optimal and safety-critical control for high-dimensional systems by combining a comparison-theorem–driven dimensionality reduction with physics-informed neural networks to solve reduced PDEs for the value function and long-horizon safety probability. The key idea is to identify low-dimensional feature coordinates that exactly capture the evolution of the cost and barrier quantities, transform the problem into low-dimensional PDEs via the Feynman-Kac framework, and solve these PDEs with PINNs that integrate data and physics. A dedicated autoencoder-like network automatically discovers meaningful features that satisfy the required bounding assumptions, enabling scalable generalization to unseen regions and longer horizons. Empirical results on synthetic high-dimensional systems show substantially better sample efficiency than path-integral Monte Carlo and successful feature learning, underscoring the approach’s potential for real-world high-dimensional control and risk quantification tasks.

Abstract

Optimal and safety-critical control are fundamental problems for stochastic systems, and are widely considered in real-world scenarios such as robotic manipulation and autonomous driving. In this paper, we consider the problem of efficiently finding optimal and safe control for high-dimensional systems. Specifically, we propose to use dimensionality reduction techniques from a comparison theorem for stochastic differential equations together with a generalizable physics-informed neural network to estimate the optimal value function and the safety probability of the system. The proposed framework results in substantial sample efficiency improvement compared to existing methods. We further develop an autoencoder-like neural network to automatically identify the low-dimensional features of the system to enhance the ease of design for system integration. We also provide experiments and quantitative analysis to validate the efficacy of the proposed method. Source code is available at https://github.com/jacobwang925/path-integral-PINN.
Paper Structure (33 sections, 1 theorem, 62 equations, 10 figures, 5 tables)

This paper contains 33 sections, 1 theorem, 62 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Path Integral Optimal Control
  4. Risk Quantification for Safe Control
  5. Physics-informed Neural Networks
  6. Problem Formulation
  7. System Dynamics
  8. Stochastic Optimal Control
  9. Safety-critical Control
  10. Proposed Method
  11. Comparison Theorem for Feature Identification
  12. PDE for Value Function with Feynman-Kac
  13. PDE for Safety Probability
  14. Physic-informed Learning
  15. Generalization: Arbitrary Feature Dimension
  16. ...and 18 more sections

Key Result

Theorem 4

Given Assumptions asm:match_upper_lower_bound and asm:continuity-condition-on-a-and-b hold, $p(x_t)$ with $x_t$ being sampled from system eq:continuous_dynamics is characterized by the following stochastic process with $\xi_0 = p(x_0)$, and $\tilde{B}_t$ being a one-dimensional standard Wiener process.

Figures (10)

  • Figure 1: The overall diagram of the proposed method for stochastic optimal value function and safety probability estimation.
  • Figure 2: The procedure diagram of the proposed method.
  • Figure 3: (\ref{['fig:PINN diagram']}) The training scheme of the physics-informed neural network (PINN). (\ref{['fig:AE diagram']}) Autoencoder-like network architecture.
  • Figure 4: Estimation of the exponential of value function at $t=0.5$ for the 1000-dimensional system by path integral MC (left), and by the proposed method (right).
  • Figure 5: Percentage error of the estimated value function with path integral MC.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Definition 1: Safety probability
  • Theorem 4
  • Remark 5
  • Remark 8
  • proof