用稳定约束的冻结卷积自动编码器，从观测数据中同时还原物理参数和场变量。
A Stability-Aware Frozen Euler Autoencoder for Physics-Informed Tracking in Continuum Mechanics (SAFE-PIT-CM)
- 在隐空间中嵌入冻结卷积作为可微分的偏微分方程求解器
- 即使采样间隔大于模拟步长，仍能准确恢复扩散系数与物理场
- 无需标签数据，适用于任意可卷积离散化的偏微分方程
材料参数如热扩散率决定微观结构在加工过程中的演化，但难以直接测量。本文提出一种基于稳定性的冻结欧拉物理信息追踪方法（SAFE-PIT-CM），其自动编码器在隐空间转移中嵌入一个冻结卷积层作为可微分的偏微分方程求解器，联合从时序观测中恢复扩散系数与底层物理场。当时间快照采样间隔超过模拟时间步长时，单步前向欧拉法违反冯·诺依曼稳定性条件，导致学习到的系数坍缩至非物理解。通过安全子步（SAFE）恢复稳定性，每个子步仅需一次冻结卷积，成本极低，且恢复误差随子步数单调收敛。在金属热扩散问题上验证，该方法无论有无预训练，均能以近完美精度同时恢复扩散系数与物理场。通过冻结算子的反向传播监督基于注意力的参数估计器，无需标签数据。该架构可推广至任何具有卷积有限差分离散化的偏微分方程。
原文摘要 · Abstract (English)
Material parameters such as thermal diffusivity govern how microstructural fields evolve during processing, but difficult to measure directly. The Stability-Aware Frozen Euler Physics-Informed Tracking for Continuum Mechanics (SAFE-PIT-CM), is an autoencoder that embeds a frozen convolutional layer as a differentiable PDE solver in its latent-space transition to jointly recover diffusion coefficients and the underlying physical field from temporal observations. When temporal snapshots are saved at intervals coarser than the simulation time step, a single forward Euler step violates the von Neumann stability condition, forcing the learned coefficient to collapse to an unphysical value. Sub-stepping with SAFE restores stability at negligible cost each sub-step is a single frozen convolution, far cheaper than processing more frames with recovery error converging monotonically with substep count. Validated on thermal diffusion in metals, the method recovers both the diffusion coefficient and the physical field with near-perfect accuracy, both with and yet without pre-training. Backpropagation through the frozen operator supervises an attention-based parameter estimator without labelled data. The architecture generalises to any PDE with a convolutional finite-difference discretisation.
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