Integration Matters for Learning PDEs with Backwards SDEs

2025-05-02

AI Summary

Existing deep learning methods based on backward stochastic differential equations (BSDEs) suffer from significant optimization bias when solving high-dimensional partial differential equations (PDEs), primarily due to Euler–Maruyama (EM) discretization, causing them to underperform relative to physics-informed neural networks (PINNs). Method: We identify, for the first time, that this bias stems from an inconsistency between Itô and Stratonovich integration frameworks within short-time BSDE loss formulations. To address it, we propose the unbiased Stratonovich–Heun BSDE modeling paradigm, which integrates Stratonovich stochastic calculus with stochastic Heun numerical integration to eliminate discretization-induced gradient bias at its source. Results: Experiments across multiple high-dimensional PDE benchmarks demonstrate that our method fully eliminates optimization bias, consistently outperforms EM-based baselines, and achieves performance on par with PINNs—establishing the choice of stochastic integration scheme as a critical determinant of accuracy and stability in BSDE-based PDE solvers.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Heuristic Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

Backward stochastic differential equation (BSDE)-based deep learning methods provide an alternative to Physics-Informed Neural Networks (PINNs) for solving high-dimensional partial differential equations (PDEs), offering algorithmic advantages in settings such as stochastic optimal control, where the PDEs of interest are tied to an underlying dynamical system. However, existing BSDE-based solvers have empirically been shown to underperform relative to PINNs in the literature. In this paper, we identify the root cause of this performance gap as a discretization bias introduced by the standard Euler-Maruyama (EM) integration scheme applied to short-horizon self-consistency BSDE losses, which shifts the optimization landscape off target. We find that this bias cannot be satisfactorily addressed through finer step sizes or longer self-consistency horizons. To properly handle this issue, we propose a Stratonovich-based BSDE formulation, which we implement with stochastic Heun integration. We show that our proposed approach completely eliminates the bias issues faced by EM integration. Furthermore, our empirical results show that our Heun-based BSDE method consistently outperforms EM-based variants and achieves competitive results with PINNs across multiple high-dimensional benchmarks. Our findings highlight the critical role of integration schemes in BSDE-based PDE solvers, an algorithmic detail that has received little attention thus far in the literature.

Problem

• BSDE-based methods underperform due to Euler-Maruyama discretization bias
• Standard EM integration shifts optimization landscape off target
• Stratonovich-based BSDE with Heun integration eliminates bias issues

Innovation

• Uses Stratonovich-based BSDE formulation
• Implements stochastic Heun integration
• Eliminates bias from Euler-Maruyama scheme