# Markus Lange-Hegermann: *Probability Distributions on Linear PDE Systems*

Differential algebra handles differential equations through symbolic manipulation, often sidestepping explicit computation or approximation of solutions. We leverage the tools of this field to equip the solution set of such PDE systems with a computationally convenient Gaussian process probability distribution. In particular, we can condition this probability distribution on (potentially noisy) data or pointwise-defined initial and boundary conditions. By operating directly with a probability distribution over the solution set of the PDE system, we streamline the treatment of systems lacking unique solutions while enabling rigorous uncertainty quantification. Notably, our approach adopts an algebraic perspective, treating PDEs uniformly without differentiation based on their type, be it elliptic, hyperbolic, or of a specific order. We showcase the efficacy of our methodology across various example PDE systems, including the heat equation, wave equation, Maxwell's equations, and control systems. In numerous experimental evaluations, we demonstrate significant enhancements in both computational efficiency and precision compared to state-of-the-art PINN methods, sometimes achieving improvements by several orders of magnitude. Technically, our approach entails an algorithmic adaptation of the Ehrenpreis-Palamodov fundamental principle, which characterizes solutions of a given PDE system via a nonlinear Fourier transform. All our proposed approaches are algorithmic, universally applicable, and incorporate approximation techniques tailored for handling large datasets.