Special Session 166: Numerical methods, viscosity solutions and free boundary problems

A PINN Framework for Identification and Control of Nonlinear PDEs with Incomplete Data

Elisa Calzola University of Ferrara Italy Co-Author(s):    Alessandro Alla, Giulia Bertaglia

Abstract: Physics-Informed Neural Networks (PINNs) offer a data-efficient approach to solving forward and inverse problems for nonlinear PDEs under sparse observations. In this talk, we introduce an OCP-PINN architecture for open-loop optimal control of time-dependent nonlinear evolution PDEs when initial/boundary data and physical parameters are only partially known. Using the adjoint method via Lagrange multipliers, we encode the coupled optimality system (composed of state equation, adjoint equation, and optimality condition) directly into a single neural network loss function. The framework employs two PINNs in series: the first identifies unknown parameters online from scattered data of the uncontrolled nonlinear dynamics; the second uses these parameters to simultaneously reconstruct the controlled state, the adjoint, and the optimal control. We demonstrate the approach on challenging nonlinear test cases: viscous Burgers (advection-diffusion balance), Allen--Cahn (phase-field instability), and Korteweg--de Vries (nonlinear dispersive waves). Despite severe data scarcity, the method achieves accurate parameter discovery and effective control toward target states, highlighting its robustness for nonlinear problems with incomplete information.

$L^p$-estimates for numerical schemes of Hamilton--Jacobi equations.

Fabio Camilli Univ. di Chieti Pescara Italy Co-Author(s):    Alessio Basti

Abstract: We establish $L^p$ error estimates for monotone numerical schemes approximating Hamilton--Jacobi equations on the $d$-dimensional torus. Using the adjoint method, we first prove a $L^1$ error bound of order one for finite-difference and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian. By interpolation, we also obtain $L^p$ estimates for every finite $p>1$. Our analysis covers a broad class of schemes, improves several existing results, and provides a unified framework for discrete error estimates.

High order Tensor-Train-based schemes for high-dimensional Mean Field Games

Elisabetta Carlini Sapienza University of Rome Italy Co-Author(s):    Luca Saluzzi

Abstract: We present a numerical method for solving high-dimensional Mean Field Games systems. The approach combines semi-Lagrangian time discretizations with Tensor-Train decompositions to mitigate the curse of dimensionality. A smoothed policy iteration algorithm is approximated using first- and second-order schemes in time. The resulting method reduces storage and computational costs from exponential to polynomial growth in the dimension. Numerical experiments illustrate the accuracy, robustness, and scalability of the proposed approach.

Convergent Numerical approximation for fractional Mean Field Games

Indranil Chowdhury Indian Institute of Technology Kanpur India Co-Author(s):

Abstract: We construct numerical approximations for fractional Mean Field Games, a coupled forward-backward system of nonlinear integro-differential equations involving Hamilton--Jacobi--Bellman and Fokker--Planck equations, where the diffusion is given by the fractional Laplacian. The method is based on finite differences and powers of the discrete Laplacian. The approximation is monotone, stable, and consistent. We discuss the convergence results of the numerical approximation to the given system.

A convergent discretization of the porous medium equation with fractional pressure

F\`elix del Teso Universidad Aut\`onoma de Madrid Spain Co-Author(s):

Abstract: We carefully construct and prove convergence of a numerical discretization of the porous medium equation with fractional pressure, \begin{equation}\label{FPE} \frac{\partial u}{\partial t} - \nabla \cdot \left( u^{m-1} \nabla (-\Delta)^{-\sigma}u \right) = 0, \end{equation} for $\sigma \in (0,1)$ and $m \geq 2$. The model, introduced by Caffarelli and V\`azquez in 2011, is currently one of two main nonlocal extensions of the local porous medium equation. It has finite speed of propagation, but as opposed to the other extension, it does not satisfy the comparison principle. We exploit the fact that the \emph{cumulative density} $v(x,t) = \int_{-\infty}^x u(y,t)\,dy$ satisfies \begin{equation*} \frac{\partial v}{\partial t} + |\partial_x v|^{m-1} (-\Delta)^s v = 0, \quad s = 1 - \sigma, \end{equation*} which is a nonlocal quasilinear parabolic equation in non-divergence form that can be analyzed through viscosity solution methods. The numerical method consists in discretizing this equation with a difference quadrature scheme with upwinding ideas and then computing the solution $u$ of \eqref{FPE} via numerical differentiation. Our results cover both absolutely continuous and Dirac or point mass initial data, and in the latter case, machinery for discontinuous viscosity solutions is needed in the analysis.

Evolving Clusters: A Dynamical Perspective on Time-Dependent Mixture Models

Adriano Festa Politecnico di Torino Italy Co-Author(s):    Adriano Festa, Fabio Camilli, Alessio Basti

Abstract: We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Unlike traditional static or heuristic approaches, we recast the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by variational cost functional rather than predefined statistical shapes, this continuous-time formulation naturally accommodates non-parametric cluster evolution. To analytically validate our general framework, we analyze the specific setting of time-dependent Gaussian mixtures. We prove that the MFG dynamics explicitly recover the trajectories of the classic Expectation-Maximization (EM) algorithm, providing a rigorous generalization that guarantees mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to smooth short-term fluctuations. Numerical experiments confirm the validity of our approach in parametric contexts and pave the way for fully non-parametric clustering applications where classical EM methods are not applicable.

Ranking Mean-Field Planning Games

Diogo Gomes KAUST Saudi Arabia Co-Author(s):    Ali Almadeh, Melih Ucer

Abstract: This paper investigates a one-dimensional Mean-Field Planning (MFP) system characterized by non-local, rank-based interactions, where individual costs depend on an agent`s relative standing within the population distribution. Using a potential formulation, we reduce the coupled system to a scalar partial differential equation and establish a rigorous equivalence between classical solutions of the ranking MFP and the associated potential problem, including explicit reconstruction formulas. By identifying a monotonicity structure within the associated operator, we prove the uniqueness of classical solutions under strict convexity assumptions. Furthermore, we address the existence of solutions in low-regularity regimes by formulating a relaxed variational inequality. Using a q-Laplacian regularization and Minty`s method, we establish the existence of weak solutions in the space of functions of bounded variation (BV). These results provide a mathematical framework for deterministic first-order planning problems with cumulative distribution couplings, such as those arising in competitive models of emission regulation.

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Robin Lien Norwegian University of Science and Technology (NTNU) Norway Co-Author(s):

Abstract: Fractional fully nonlinear PDEs such as Hamilton-Jacobi-Bellman and Isaacs equations arise naturally in optimal control and differential game theory, with many applications in engineering, science, economics, etc. We study discretizations of such equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

From the Monge-Amp\`ere equation to Stochastic Optimal Control and Reinforcement Learning

Ivan Majic UCL England Co-Author(s):    Dr Max Jensen

Abstract: The Monge--Amp\`ere equation is a nonlinear second-order partial differential equation involving the determinant of the Hessian of an unknown function, with applications in areas such as Optimal Transport. Under appropriate conditions, the Monge--Amp\`ere equation has an equivalent Hamilton--Jacobi--Bellman (HJB) formulation in the classical and viscosity solution sense. This talk studies a model Monge--Amp\`ere equation through its associated HJB formulation and the corresponding Stochastic Optimal Control problem. We establish the existence of strong controls for the control problem, and use a verification theorem to connect the value function to the HJB equation and hence to the original Monge--Amp\`ere equation. Finally, we will showcase examples of using Reinforcement Learning to solve the problem numerically.

Rates of convergence of finite element approximations of second-order mean field games with nondifferentiable Hamiltonians

Iain Smears University College London England Co-Author(s):    Yohance A. P. Osborne

Abstract: We consider the analysis of numerical methods for second-order mean field games. In the setting of nondifferentiable Hamiltonians, the system comprises the HJB equation for the value function and the KFP partial differential inclusion for the density of the players. In the nondifferentiable setting, it is known from examples that one cannot expect any quantitative control on the error made in approximating the unknown drift term that appears in the differential inclusion. Despite this difficulty, we prove a rate of convergence for finite element approximations for such MFG systems, on general bounded polytopal Lipschitz domains with strongly monotone running costs. In particular, we obtain a rate of convergence in the $H^1$-norm for the value function approximations and in the $L^2$-norm for the approximations of the density.

A numerical approach to degenerate fully nonlinear elliptic problems from the pure equation to free boundaries

Ercilia Sousa University of Coimbra Portugal Co-Author(s):    Edgard Pimentel

Abstract: We study a class of degenerate fully nonlinear elliptic equations, consisting of a pure equation and an associated transmission-type free boundary problem. First, we propose a regularization of the pure equation and develop a numerical method for the resulting regularized problem. This regularization also plays a role in ensuring the monotonicity of the numerical scheme. We prove that the method is monotone, consistent, and stable. Consequently, the Barles-Souganidis framework guarantees the convergence of the numerical approximation. Once the numerical treatment of the pure equation is established, we show how a similar approach can be extended to the free boundary transmission problem. Finally, we present numerical experiments that support the theoretical results. This work has been done in collaboration with Edgard Pimentel.