GENERIC动力学的保结构不确定性量化

中文摘要 AI 辅助

保结构机器学习将物理结构直接嵌入模型架构，但这类带硬约束模型的不确定性量化（UQ）仍存在局限，因为标准UQ方法可能违反编码的可容许性条件、需要修改架构或带来大量计算成本。本研究提出保结构认知神经网络（S-PENNs），这是面向带硬架构约束的科学机器学习模型的通用UQ框架，并将其实例化用于GENERIC（非平衡可逆-不可逆耦合的通用方程）动力学。S-PENNs通过在预训练模型的受约束组件上附加轻量级认知网络（epinets）来保留模型的结构约束，确保每个采样实现从构造上始终符合物理可容许性。当应用于GENERIC动力学时，该框架生成符合热力学一致性的滚动预测，同时保留热力学第一和第二定律。此外，本研究将S-PENNs与拆分共形预测作为事后校准方法结合，以生成具有有限样本边际覆盖率保证的预测区间。本研究在三个数值算例上验证了S-PENNs：由常微分方程（ODE）描述的耦合热浴的简谐振子、理想化化学马达，以及由偏微分方程（PDE）描述的一维黏塑性模型。在所有三个算例中，S-PENNs均生成符合热力学一致性的随机实现和校准良好的预测区间，且与深度集成相比计算成本降低约1至3个数量级。尽管本研究聚焦于GENERIC动力学，但S-PENNs可更广泛地扩展到计算力学中带硬或软约束的科学机器学习模型。

英文摘要

Structure-preserving machine learning embeds physical structure directly into model architectures, yet uncertainty quantification (UQ) for such hard-constrained models remains limited because standard UQ methods may violate the encoded admissibility conditions, require architectural modifications, or impose substantial computational costs. In this work, we propose Structure-Preserving Epistemic Neural Networks (S-PENNs), a general framework for UQ in scientific machine learning models with hard architectural constraints, and instantiate it for GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) dynamics. S-PENNs preserve the structural constraints of a pretrained model by attaching lightweight epinets to its constrained components, ensuring that every sampled realization remains physically admissible by construction. When applied to GENERIC dynamics, such a proposed framework yields thermodynamically consistent rollouts that preserve the first and second laws. Furthermore, we combine S-PENNs with split conformal prediction as a post-hoc calibration method to produce prediction intervals with finite-sample marginal coverage guarantees. We validate S-PENNs on three numerical examples: a harmonic oscillator coupled to a heat bath and an idealized chemical motor, both governed by ODEs, and a one-dimensional viscoplastic model governed by PDEs. Across all three examples, S-PENNs produce thermodynamically consistent stochastic realizations and well-calibrated prediction intervals while reducing the computational cost by about one to three orders of magnitude compared to deep ensembles. Although the present study focuses on GENERIC dynamics, S-PENNs can be extended more broadly to scientific machine learning models in computational mechanics with either hard or soft constraints.

发表机构

• University of Pennsylvania（宾夕法尼亚大学）

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