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Physics-Informed Representation and Learning: Control and Risk Quantification
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Proposes comparison-theorem dimensionality reduction plus a physics-informed neural network (PINN) to estimate value functions and safety probabilities of high-dimensional stochastic systems via low-dimensional PDEs, with an autoencoder-like network for automatic feature identification.
Problem
Efficient optimal and safety-probability estimation for high-dimensional stochastic control systems, where Monte Carlo and path-integral methods scale poorly with state dimension and horizon.
Approach
Use a comparison theorem to reduce a high-dimensional stochastic system to low-dimensional scalar SDE features whose Feynman-Kac/hitting-time PDEs are solved by a PINN, with an autoencoder-like network to learn the features automatically.
Key results
- PINN training takes about 80 s versus roughly 3000 s for path integral Monte Carlo sampling on the 1000-dimensional value function task.
- The autoencoder-like network identifies the analytically derived features with MSE 0.2 and 0.06 for the two features of the 3-dimensional system.
- PINN prediction error stays below 0.08 absolute error for t from 0.5 to 1.5, generalizing to unseen time horizons.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the reduced PDE and PINN instead of path integral Monte Carlo | Value function estimation · 1000-dimensional system | computation time: 80s v.s ~3000s | Helped |
| Train for longer, up to 50000 epochs (ablation) | PINN · Value function estimation | Prediction Error (%): 7.104 -> 5.322 -> 5.475 -> 5.940 -> 4.733 | Helped |
| Increase training beyond 50000 epochs (ablation) | PINN · Value function estimation | Prediction Error (%): 4.733 -> 4.809 | No clear effect |
| Increase PDE constraint grid points (ablation) | PINN · Value function estimation | Prediction Error (%): 5.498 -> 5.391 -> 5.749 -> 5.927 -> 5.478 -> 5.283 -> 5.153 -> 4.992 | Helped |
| Use a deeper autoencoder architecture (ablation) | Autoencoder-like feature learner · 3-dimensional system feature learning | Representation Error (%): 0.248 -> 0.112 -> 0.007 | Helped |
| Increase units per layer in the 1-layer autoencoder (ablation) | Autoencoder-like feature learner · 3-dimensional system feature learning | Cost Reconstruction Error (%): 0.901 -> 0.984 -> 1.003 | Hurt |
| Use the 5-layer autoencoder architectures (ablation) | Autoencoder-like feature learner · 3-dimensional system feature learning | Cost Reconstruction Error (%): 0.120, 0.099, 0.129 | Helped |
Limitations
- Method requires the high-dimensional system to admit a low-dimensional representation of the value function or safety probability (Assumptions on comparison theorem), which is a necessary condition.
- Experiments are limited to linear or simple synthetic systems with 1000-dimensional and 3-dimensional examples.
- Feature learning error remains non-trivial (MSE 0.2 for feature 1) and preimage computation relies on grid search that becomes expensive in larger systems.
Experimental setup
Datasets: synthetic data generated from PDE/Monte Carlo simulation of linear stochastic systems
Benchmarks: 1000-dimensional linear stochastic system value function estimation 3-dimensional system sample complexity analysis 3-dimensional safety probability estimation
Metrics: percentage error absolute error MSE of learned features computation time
Baselines: path integral Monte Carlo standard Monte Carlo safety probability estimation
Builds on: path integral control physics-informed neural networks comparison theorem for stochastic differential equations Feynman-Kac formula
Closest papers by meaning
- Physics-Informed Neural Network Policy Iteration: Algorithms, Convergence, and Verification (ICML 2024 · similarity 0.88)
- Safely Learning Controlled Stochastic Dynamics (NeurIPS 2025 · similarity 0.87)
- S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks (NeurIPS 2026 · similarity 0.86)
- A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems (ICML 2025 · similarity 0.86)
- Learning Reduced Fluid Dynamics (AAAI 2024 · similarity 0.84)
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.84)
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