Yumiharu Nakano

Yumiharu Nakano

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

Yumiharu Nakano

We propose numerical integration methods for Choquet integrals where the capacities are given by distortion functions of an underlying probability measure. It relies on the explicit representation of the integrals for step functions and can be seen as quasi-Monte Carlo methods in this framework. We give bounds on the approximation errors in terms of the modulus of continuity of the integrand and the star discrepancy.

Yumiharu Nakano

This paper studies convergence of differentiable approximation schemes for terminal value problems of fully nonlinear parabolic partial differential equations. Differentiable schemes approximate spatial derivatives directly via gradients of smooth basis functions and are therefore generally non-monotone, placing them outside the scope of the classical Barles--Souganidis convergence theory. To address this, we introduce an abstract framework for differentiable schemes and establish convergence under conditions we call approximate monotonicity and weak stability. A central tool is a max-min representation of the nonlinearity, which allows us to relax the strict monotonicity condition when the approximate solution is smooth. We apply the abstract framework to kernel-based function approximation methods and derive qualitative convergence for general equations and quantitative error estimates for Hamilton--Jacobi--Bellman equations. Numerical experiments support the theoretical results and confirm the computational feasibility of the proposed approach.

Yumiharu Nakano

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the stochastic Nelson problem.