用符号学习方法解析随机微分方程的确定性部分，提升可解释性与泛化能力。
Identifying Unknown Stochastic Dynamics via Finite expression methods
- 提出有限表达法（FEX），通过符号学习提取确定性项的数学表达式。
- 在多维、非线性系统中，比神经网络模型更准确且外推能力强。
- 结果具有科学可解释性，适合需要机制发现的科研场景。
建模随机微分方程（SDE）对于理解各类科学领域的复杂动力系统至关重要。现有方法多采用基于神经网络的模型，通常将SDE表示为确定性与随机项的组合，但这类模型缺乏可解释性，且难以超越训练域泛化。本文提出有限表达法（FEX），一种符号学习方法，用于推导SDE中确定性部分的可解释数学表达式。对随机部分，将FEX与先进生成建模技术结合，实现对SDE的完整表征。在低维、高维及非线性SDE上的数值实验表明，FEX在训练域外具有优异泛化能力，长期预测精度优于神经网络方法。FEX识别出的符号表达式不仅提升预测准确性，还揭示了系统底层动力学机制，为科学发现提供新路径。
原文摘要 · Abstract (English)
Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty generalizing beyond their training domain. This paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems, paving the way for new scientific discoveries.
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