The Segal-Bargmann Transform in Neural Partial Differential Equations

Abstract

We introduce a novel neural operator method based on Segal-Bargmann transform and Bargmann-Fock space for learning the solution operators of partial differential equations (PDEs). The method brings data encoding, latent-space representation, operator learning, and decoding into an unifed framework. From only finitely many observations, we establish a sup-norm reconstruction error of order $ L^{-\frac{1}{2}} $, where L is the number of complex sampling points that are decoupled with the original samplings and observations. The reconstruction error is independent of the spatial dimension and the number of original sampling points. We show that this new mesh-free Segal-Bargmann Neural Operator approach has superior performance compared to existing NO approaches like FNO, BelNet, GNOT, etc.,\ particularly in terms of pointwise control $ \sup_{x}\lvert\hat{u}_{NO} - u_{truth}\rvert $ for solutions of PDEs $ \mathcal{L}u = f $ which provides pointwise accuracy and interpretability as classical numerical solvers. We also demonstrates the reconstructions of the solution operators for higher dimensional nonlinear PDEs, which suggest that our proposed method has great potential to be applied to more challenging PDEs and to even higher spatial dimensions.