Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent. As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime. Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.