Physics-Informed Neural Networks Learn PDE Solution Families.

Solving Partial Differential Equations (PDEs) is fundamental across many scientific and engineering disciplines. This paper presents a novel physics-informed framework for learning compact, finite-dimensional representations, or "embeddings," of entire families of PDE solutions. The approach utilizes a multihead Physics-Informed Neural Network (PINN) architecture. In this architecture, a shared neural network body learns a latent manifold that effectively represents the underlying solution space of the PDE. Simultaneously, separate linear "heads" are used to reconstruct individual solutions, each corresponding to a different initial condition. A crucial component is an orthogonalization penalty applied to the heads, which helps remove redundancies in the latent representation and stabilizes the principal-component spectrum during training. The method was successfully applied to the one-dimensional viscous Burgers equation, with robustness checks on heat and wave equations. For instance, with a latent dimension of 20, the learned manifolds for Burgers dynamics showed significant effective dimensional reduction, with only 2-4 principal components capturing about 95% of the latent-space variance. The resulting frequency profiles and principal components are robust and reproducible across independent training runs, providing valuable insights into the geometry of the solution manifold.