  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.
