深度学习能否实现跨物理映射？

中文摘要 AI 辅助

深度学习能否翻译由根本不同方程支配的物理场？我们通过引入跨物理映射（CPM）来解决这个问题，这是一个用于异构物理域之间映射的算子学习框架。我们通过兼容的潜在表示来阐述此类映射的充分条件，并提出一个无量纲缩放原理，该原理在不假设源系统和目标系统动力学等价的情况下，对齐其特征演化尺度。作为代表性测试，成对的扩散场和波场分别从各自的抛物型和双曲型方程独立生成，同时共享相同的潜在几何、材料非均匀性、激励和无量纲尺度。我们评估了七种架构——ResUNet、DeepONet、傅里叶、潜在、小波、U形和Galerkin神经算子——用于扩散到波和波到扩散两种映射。结果揭示了强烈的方向不对称性。扩散到波的重建更具挑战性，因为它需要恢复被扩散衰减的波前、相位和飞行时间信息；U-NO在此方向上表现最佳，实现了相对$\ell_2$误差0.307和$R^2$为0.905。波到扩散映射则相当稳定，GNO实现了相对$\ell_2$误差0.154和$R^2$为0.935。神经算子通常优于传统卷积基线，突显了跨物理变换的非局部性质。这些发现表明，深度学习可以在共享潜在流形上建立不同物理模态之间的有用映射，而可实现的精度仍然从根本上受制于控制物理的与方向相关的信息内容。

英文摘要

Can deep learning translate physical fields governed by fundamentally different equations? We address this question by introducing Cross-Physics Mapping (CPM), an operator-learning framework for mappings between heterogeneous physical domains. We formulate sufficient conditions for such mappings through compatible latent representations and propose a dimensionless scaling principle that aligns the characteristic evolution scales of the source and target systems without assuming their dynamical equivalence. As a representative test, paired diffusion and wave fields are generated independently from their respective parabolic and hyperbolic equations while sharing the same latent geometry, material heterogeneity, excitation, and dimensionless scale. Seven architectures-ResUNet, DeepONet, Fourier, latent, wavelet, U-shaped, and Galerkin neural operators-are evaluated for both diffusion-to-wave and wave-to-diffusion mappings. The results reveal a strong directional asymmetry. Diffusion-to-wave reconstruction is more challenging because it requires recovering wavefront, phase, and time-of-flight information attenuated by diffusion; U-NO performs best in this direction, achieving a relative $\ell_2$ error of $0.307$ and an $R^2$ of $0.905$. Wave-to-diffusion mapping is considerably more stable, with GNO attaining a relative $\ell_2$ error of $0.154$ and an $R^2$ of $0.935$. Neural operators generally outperform the conventional convolutional baseline, highlighting the nonlocal nature of cross-physics transformations. These findings demonstrate that deep learning can establish useful mappings between distinct physical modalities on a shared latent manifold, while the achievable accuracy remains fundamentally constrained by the direction-dependent information content of the governing physics.

发表机构

• Bundesanstalt für Materialforschung and -prüfung (BAM)（德国联邦材料研究与测试研究所）

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