Learning embeddings of non-linear PDEs: the Burgers' equation

• URL: http://arxiv.org/abs/2603.07812v1
• Date: Sun, 08 Mar 2026 21:32:31 GMT
• Title: Learning embeddings of non-linear PDEs: the Burgers' equation
• Abstract summary: We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup.<n>We extract non-degenerate information from them using principal component analysis (PCA)<n>A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations.
• Score: 0.6474760227870044
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup, and extract non-degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal-component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.

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