Verifier-Guided Model Discovery for Physical Dynamical Systems with Pretrained Symbolic Transformers

基于预训练符号变换器的物理动力系统验证器引导模型发现

Abstract: Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained transformers mapping synthetic ODE trajectories to equations offer interpretable alternatives, promising transfer without system-specific equation knowledge. Transferring them reliably to high-dimensional physical data, however, remains an open challenge.

摘要： 对非线性物理系统进行可靠的预测是科学发现和工程决策的基础。然而，高保真模拟的成本极其高昂，而机器学习代理模型往往具有不透明性，且编码了关于系统动力学的假设，从而限制了其泛化能力。将合成常微分方程（ODE）轨迹映射为方程的预训练变换器（Transformers）提供了一种可解释的替代方案，有望在无需特定系统方程知识的情况下实现迁移。然而，如何将这些模型可靠地迁移到高维物理数据中，仍然是一个亟待解决的挑战。

We develop a verifier-guided (VG) workflow around ODEFormer as a symbolic backbone, using dynamical and physical-admissibility criteria to select from a multi-trajectory candidate equation pool, enabling transfer. On canonical Van der Pol oscillators, VG outperforms the original ODEFormer workflow across held-out initial conditions. We then address vortex shedding, a phenomenon occurring in atmospheric and plasma systems of societal relevance, through coordinate reduction and symbolic discovery at fixed and varying Reynolds numbers.

我们开发了一种以 ODEFormer 为符号主干的验证器引导（VG）工作流，利用动力学和物理可采纳性准则从多轨迹候选方程池中进行筛选，从而实现模型迁移。在经典的范德波尔（Van der Pol）振荡器上，VG 在留出的初始条件下表现优于原始的 ODEFormer 工作流。随后，我们通过坐标约简和在固定及变化雷诺数下的符号发现，研究了大气和等离子体系统中具有社会意义的现象——涡旋脱落。

VG discovers fixed-parameter reduced-order equations that recover the fundamental shedding oscillator and higher harmonics without a wake-specific candidate library or prescribed Navier-Stokes structure, while the cross-parameter model generalizes to withheld regimes. Reconstruction fidelity alone did not determine symbolic discoverability, highlighting the importance of compatibility between latent dynamics and the backbone’s pretraining distribution. This work establishes a verifier-guided neural-to-symbolic methodology for interpretable and physically auditable forecasting in the natural sciences.

VG 发现了固定参数的降阶方程，这些方程能够恢复基本的脱落振荡器和高次谐波，且无需特定于尾流的候选库或预设的纳维-斯托克斯（Navier-Stokes）结构；同时，跨参数模型能够泛化到未见过的工况中。研究发现，仅凭重构保真度并不能决定符号的可发现性，这凸显了潜在动力学与主干模型预训练分布之间兼容性的重要性。本研究建立了一种验证器引导的神经符号方法，为自然科学领域的可解释且物理可审计的预测提供了新途径。