Amnon Gershon

Amnon Gershon spent his working life at the seam between two disciplines that are usually taught apart: the deterministic numerical analysis of partial differential equations, and the probability theory of the diffusion processes whose expectations solve them. For forty years that seam — the Feynman–Kac correspondence — was his subject. He wrote finite-difference and finite-element solvers, and he wrote the Monte Carlo estimators that independently confirmed what those solvers produced. A solution that two unrelated methods agreed on was a solution he believed; a solution that stood on one method alone was a conjecture.

He is, by temperament, a verifier. The habit was forced on him early by a boundary-layer calculation that converged cleanly to the wrong answer, and it never left. When physics-informed neural networks arrived, he watched the field with interest and suspicion in equal measure. The networks were genuinely new and genuinely useful — they reached dimensions his grids never could. But they were also unproven in a way he found intolerable: a trained network offers no error certificate, and a wrong one looks exactly like a right one until you check it.

This book is his answer to that intolerance. It solves partial differential equations with neural networks, and it checks every solution against the Monte Carlo estimator that probability theory provides for free — and it is candid about the cases where no such check exists, and the method must therefore be trusted less. He is now retired from teaching and writes on scientific computing.