透過具可解釋自動微分校正的領域自適應物理資訊神經網路修正模型錯誤設定

Abstract

物理資訊神經網路（PINNs）已成為模擬複雜動態系統的強大工具，透過將物理定律（以微分方程形式表示）融入神經網路架構中。然而，其效能受到模型錯誤設定的顯著限制，當物理先驗知識不完整或錯誤時，會導致非物理的解以及預測精度的下降。為了解決此挑戰，我們提出領域自適應物理資訊神經網路（DAPINNs）框架，並結合基於自動微分的物理校正（ADPC）模型。此框架透過三階段流程整合部分物理知識與資料驅動的校正：源域預訓練、目標域與ADPC的微調，以及用於差異識別的符號迴歸。ADPC模型利用自動微分技術動態校正錯誤設定的控制方程，捕捉包括高階與非線性交互作用在內的複雜物理現象。交替更新策略提升了訓練穩定性，而符號迴歸確保校正結果的可解釋性，從而增進科學理解。透過結合領域自適應與穩健的校正機制，DAPINNs與ADPC提供了一個多功能且具可解釋性的解決方案，適用於在不完整物理知識下模擬動態系統。

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for modeling complex dynamical systems by embedding physical laws, expressed as differential equations, into neural network architectures. However, their performance is critically limited by model misspecification, where incomplete or incorrect physical priors lead to non-physical solutions and diminished predictive accuracy. To address this challenge, we propose the Domain-Adaptive Physics-Informed Neural Networks (DAPINNs) framework augmented with an Auto-Differentiation-based Physics Correction (ADPC) model. This framework integrates partial physical knowledge with data-driven corrections through a three-stage pipeline: source-domain pre-training, target-domain fine-tuning with ADPC, and symbolic regression for discrepancy identification. The ADPC model leverages automatic differentiation to dynamically correct misspecified governing equations, capturing complex physical phenomena, including higher-order and nonlinear interactions. An alternating update scheme enhances training stability, while symbolic regression ensures interpretable corrections, improving scientific understanding. By combining domain adaptation with robust correction mechanisms, DAPINNs with ADPC offers a versatile and interpretable solution for modeling dynamical systems under incomplete physical knowledge.