Physics-Informed Neural Embeddings of PDE Solution Families

• URL: http://arxiv.org/abs/2607.06348v1
• Date: Tue, 07 Jul 2026 14:45:42 GMT
• Title: Physics-Informed Neural Embeddings of PDE Solution Families
• Abstract summary: We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations.<n>The method uses a multihead Physics-Informed Network in which a shared body learns a latent manifold representing the solution space.<n>A head-orthogonalization penalty removes viscous degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations.
• Score: 0.6575205226217063
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.

Related papers

• Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations [6.85637056992485]
  Solving partial differential equations with high-frequency solutions remains a central challenge in physics-informed machine learning.<n>This paper proposes a Frequency Shift Physics-Informed Extreme Learning Machine framework that addresses this limitation.<n>Two variants are developed: FS-PIELM-L assigns independent frequency magnitudes to individual neurons, while FS-PIELM-G groups neurons for improved robustness.
  arXiv  Detail & Related papers  (2026-07-02T04:37:40Z)
• The Hamilton-Jacobi Theory of Deep Learning [0.0]
  Training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems.<n>The correspondence is exact for log-sum-exp layers and structural for broader architectures.
  arXiv  Detail & Related papers  (2026-05-27T18:38:23Z)
• Escape dynamics and implicit bias of one-pass SGD in overparameterized quadratic networks [3.198538967655537]
  We analyze the one-pass gradient descent dynamics of a two-layer neural network with quadratic activations in a teacher--student framework.<n>We study the low-dimensional ordinary differential equations that govern the evolution of the student--teacher and student--student overlap matrices.
  arXiv  Detail & Related papers  (2026-04-03T14:47:24Z)
• Learning embeddings of non-linear PDEs: the Burgers' equation [0.6474760227870044]
  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.
  arXiv  Detail & Related papers  (2026-03-08T21:32:31Z)
• On the Mechanism and Dynamics of Modular Addition: Fourier Features, Lottery Ticket, and Grokking [49.1352577985191]
  We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task.<n>Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics.
  arXiv  Detail & Related papers  (2026-02-18T20:25:13Z)
• Phase-space entropy at acquisition reflects downstream learnability [54.4100065023873]
  We propose an acquisition-level scalar $S_mathcal B$ based on instrument-resolved phase space.<n>We show theoretically that (S_mathcal B) correctly identifies the phase-space coherence of periodic sampling.<n>$|S_mathcal B|$ consistently ranks sampling geometries and predicts downstream reconstruction/recognition difficulty emphwithout training.
  arXiv  Detail & Related papers  (2025-12-22T10:03:51Z)
• Network scaling and scale-driven loss balancing for intelligent poroelastography [2.665036498336221]
  A deep learning framework is developed for multiscale characterization of poroelastic media from full waveform data. Two major challenges impede direct application of existing state-of-the-art techniques for this purpose. We propose the idea of emphnetwork scaling where the neural property maps are constructed by unit shape functions composed into a scaling layer.
  arXiv  Detail & Related papers  (2024-10-27T23:06:29Z)
• Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space [49.1574468325115]
  We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace.
  arXiv  Detail & Related papers  (2024-05-27T16:11:39Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Conditions for realizing one-point interactions from a multi-layer structure model [77.34726150561087]
  A heterostructure composed of $N$ parallel homogeneous layers is studied in the limit as their widths shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schr"odinger equation is given.
  arXiv  Detail & Related papers  (2021-12-15T22:30:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.