  We develop a provably convergent kernel-based solver for path-dependent PDEs
(PPDEs). Our numerical scheme leverages signature kernels, a recently
introduced class of kernels on path-space. Specifically, we solve an optimal
recovery problem by approximating the solution of a PPDE with an element of
minimal norm in the signature reproducing kernel Hilbert space (RKHS)
constrained to satisfy the PPDE at a finite collection of collocation paths. In
the linear case, we show that the optimisation has a unique closed-form
solution expressed in terms of signature kernel evaluations at the collocation
paths. We prove consistency of the proposed scheme, guaranteeing convergence to
the PPDE solution as the number of collocation points increases. Finally,
several numerical examples are presented, in particular in the context of
option pricing under rough volatility. Our numerical scheme constitutes a valid
alternative to the ubiquitous Monte Carlo methods.
