Deep learning maps one physics type to another despite key differences
Can Deep Learning Achieve Cross-Physics Mapping?
Machine Learning
Summary
This paper explores whether deep learning can translate patterns from one type of physical process (like heat spreading) to another quite different one (like waves moving). The authors created a method to connect these different physics by turning their data into a shared hidden form. They tested several neural network designs to map fields governed by diffusion equations into wave equations and back. The results show that going from diffusion to waves is harder because some details are lost, but going from waves to diffusion works better. Overall, deep learning can link these different physical systems but some limits depend on the physics involved.
What this means in practice
- •For engineering simulation teams: Translate simulation results from diffusion-based models to wave-based models to accelerate hybrid physics analyses.
- •For geophysics modelers: Improve interpretation of subsurface wave data by mapping diffusion-type property distributions through learned transformations.
Authors
Pengfei Zhu, Julien Lecompagnon, Mathias Ziegler
Abstract
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.