LEARNING EMBEDDINGS OF NON-LINEAR PDES: THE BURGERS’ EQUATION

Raul Jimenez, Pavlos Protopapas, Leonid Sarieddine, Pedro Tarancón-Álvarez

Keywords: Physics-Informed Neural Networks, Multi-head architectures, Embeddings, Principal Component Analysis

TL;DR: A multi-head PINN learns a shared latent representation of Burgers’ equation whose orthogonalized PCA decomposition offers a stable diagnostic of effective dimensionality across families of solutions.

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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Submission Number: 92