Juan and Phil are members of CASUS’ Young Investigator Group “Mathematical Foundations of Complex System Science” led by Michael Hecht.
In the talk, a novel finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials will be introduced. Based on this surrogate model, Juan and Phil realize a convex variational formulation, solving general partial differential equations (PDEs), and yield an alternative to physics-informed neural nets (PINNs).
In contrast to PINNs, the convexity induces exponential fast convergence to the approximate solution of the truncated problem. This gain in efficiency is complemented by an increase of approximation power, thereby, outperforming PINN alternatives in both accuracy and runtime. Apart from the empirical evidence, presented herein, the canonical translation of classic PDE theory in terms of the finite-dimensional Sobolev spaces suggests the universality of the approach.