# Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems

arXiv: https://arxiv.org/abs/2407.08868 · HTML: https://mlsanity.com/paper/2407.08868/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 74 references that are in the map.
- First posted on arXiv 11 Jul 2024.

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A Review of Safe Reinforcement Learning: Methods, Theory and Applications](https://mlsanity.com/paper/2205.10330/) (2022)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Gradients as a Measure of Uncertainty in Neural Networks](https://mlsanity.com/paper/2008.08030/) (2020)
- [Signal Processing for Implicit Neural Representations](https://mlsanity.com/paper/2210.08772/) (2022)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs](https://mlsanity.com/paper/2009.01196/) (2020)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [Safe Reinforcement Learning with Chance-constrained Model Predictive Control](https://mlsanity.com/paper/2112.13941/) (2021)
- [Accelerating Physics-Informed Neural Network Training with Prior Dictionaries](https://mlsanity.com/paper/2004.08151/) (2020)
- [Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems](https://mlsanity.com/paper/2005.04374/) (2020)
- [A Barrier Function Approach to Finite-Time Stochastic System Verification and Control](https://mlsanity.com/paper/1909.05109/) (2019)
- [Learning Gradients of Convex Functions with Monotone Gradient Networks](https://mlsanity.com/paper/2301.10862/) (2023)
- [Myopically Verifiable Probabilistic Certificate for Long-term Safety](https://mlsanity.com/paper/2110.13380/) (2021)

## Built on by (most cited first, 2 of 2)

- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Physics-informed Reinforcement Learning for Stochastic Reach-Avoid Analysis](https://mlsanity.com/paper/2608.17117/) (2026)

## Abstract (arXiv)

Accurate estimation of long-term risk is essential for the design and analysis of stochastic dynamical systems. Existing risk quantification methods typically rely on extensive datasets involving risk events observed over extended time horizons, which can be prohibitively expensive to acquire. Motivated by this gap, we propose an efficient method for learning long-term risk probabilities using short-term samples with limited occurrence of risk events. Specifically, we establish that four distinct classes of long-term risk probabilities are characterized by specific partial differential equations (PDEs). Using this characterization, we introduce a physics-informed learning framework that combines empirical data with physics information to infer risk probabilities. We then analyze the theoretical properties of this framework in terms of generalization and convergence. Through numerical experiments, we demonstrate that our framework not only generalizes effectively beyond the sampled states and time horizons but also offers additional benefits such as improved sample efficiency, rapid online inference capabilities under changing system dynamics, and stable computation of probability gradients. These results highlight how embedding PDE constraints, which contain explicit gradient terms and inform how risk probabilities depend on state, time horizon, and system parameters, improves interpolation and generalization between/beyond the available data.