Minh-Binh Tran

Avy Soffer, Nguyen Gia Hien, Minh-Binh Tran

We introduce a microlocal phase-space-filtered physics-informed neural network (PINN--TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.

Minh-Binh Tran

Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new Domain Decomposition Scheme to solve forward-backward stochastic differential equations (FBSDEs) parallely. We reconstruct the Four Step Scheme in {MaProtterYong:1994:SFB} with some different conditions and then associate it with the idea of Domain Decomposition Methods. We also introduce a new technique to prove the convergence of Domain Decomposition Methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs.

Minh-Binh Tran

We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains, and then apply it to our problem.

Minh-Binh Tran

In this paper, we partially answer open questions about the convergence of overlapping Schwarz methods. We prove that overlapping Schwarz methods with Dirichlet transmission conditions for semilinear elliptic and parabolic equations always converge. While overlapping Schwarz methods with Robin transmission conditions only converge for semilinear parabolic equations, but not for semilinear elliptic ones. We then provide some conditions so that overlapping Schwarz methods with Robin transmission conditions converge for semilinear elliptic equations. Our new techniques can also be potentially applied to others kinds of partial differential equations.

Minh-Binh Tran

We introduce in this paper a new tool to prove the convergence of the Overlapping Optimized Schwarz Methods with multisubdomains. The technique is based on some estimates of the errors on the boundaries of the overlapping strips. Our guiding example is an n-Dimensional Linear Parabolic Equation.

Maximilian Kurbanov, Minh-Nhat Phung, Minh-Binh Tran

The numerical reconstruction of controls for nonlinear partial differential equations remains a challenging and relatively underdeveloped problem, despite the extensive literature on control theory. While recent works have introduced constructive approaches for semilinear wave and heat equations, the design of reliable computational methods for approximating control functions continues to raise significant analytical and numerical difficulties. In this work, we propose a novel framework based on physics-informed neural networks (PINNs) for the approximation of controls in nonlinear PDE settings. We develop an approach that incorporates the governing equations, boundary conditions, and control mechanisms directly into the learning process. In addition, we provide a convergence analysis of the proposed method and support the theoretical findings with numerical experiments demonstrating good performance. The resulting framework offers a flexible computational tool for approximating control functions from partial observations and provides a promising direction for the computational treatment of control reconstruction problems. Moreover, it can be applied to a broader class of problems, beyond the control of nonlinear PDEs.

Alain Bensoussan, Thien P. B. Nguyen, Minh-Binh Tran et al.

We propose a splitting approach to solve the second-order Hamilton--Jacobi equation, reducing it to a heat step and a purely first-order step. The latter is implemented using a gradient value policy iteration algorithm, enabling efficient characteristic-based machine learning methods. We establish convergence rates for the splitting method. In particular, the $L^\infty$ error is bounded below by $\mathcal{O}(h)$ and above by $\mathcal{O}(h^{1/7})$ for Lipschitz initial data; this improves to $\mathcal{O}(h^{1/5})$ for semiconcave data and to $\mathcal{O}(h^{1/3})$ for $C^2$ data. We also prove an upper $L^1$ error estimate of order $\mathcal{O}(h^{1/2})$ in the periodic setting, where $h$ is the splitting step. For the first-order step, we provide a weighted $L^2$ error analysis that shows exponential convergence. Each iteration solves linear characteristic equations and learns the value function by minimizing a weighted value gradient loss. The approach yields stable and accurate numerical results.

Tan-Phuc Nguyen, Minh-Binh Tran, Son Tu

The numerical reconstruction of controls for partial differential equations remains comparatively underdeveloped, despite the extensive analytical literature on controllability. This difficulty is particularly pronounced for wave equations, whose conservative structure, oscillatory dynamics, and high-frequency behavior make direct discretization and optimization challenging. In this work, we introduce a Neural-Spectral method for approximating controls of wave equations. The method represents both the state and the control in a Dirichlet spectral basis and parameterizes the time-dependent modal coefficients using shallow neural networks. In this way, the spatial oscillatory structure of the wave equation is built into the approximation, and the learning task is reduced to reconstructing temporal coefficients. We prove approximation results showing that, under the standing assumption that an exact control exists in the relevant energy framework, the control-state pairs found can approximate exact controlled trajectories uniformly in time in the energy norm, while also approximating the corresponding controls in \(L^2\). We also state a conditional computable error estimate that separates spectral truncation, neural-network approximation, quadrature, and optimization errors. In addition, we discuss structural obstructions faced by standard time-stepping schemes for conservative wave dynamics: explicit Euler amplifies high frequencies, implicit Euler introduces artificial dissipation, and Crank--Nicolson preserves amplitudes but compresses high-frequency phases. Numerical experiments in one, two, and three space dimensions illustrate the method on nonlinear, linear-reference, and high-dimensional control benchmarks.

Cong B. Van, Thien P. B. Nguyen, Minh-Binh Tran et al.

We study an inverse initial data problem for a convection-diffusion equation with memory, where the goal is to recover the unknown initial condition from final-time data. The model includes convection, an instantaneous Laplacian term, and a nonlocal-in-time memory term involving the Laplacian of the past states, which leads to a severely ill-posed backward problem. We prove uniqueness in a spatially independent coefficient setting by applying the Fourier transform and using an analyticity argument for a scalar Volterra equation. For the variable-coefficient case, we develop a computational method based on Legendre spatial dimensional reduction and Tikhonov regularization. The solution is approximated by a finite tensor-product Legendre expansion, thereby reducing the inverse problem to a finite-dimensional terminal-value system for the time-dependent coefficients. We solve the reduced problem by a Tikhonov-regularized least-squares method with an $H^2$ penalty. For a fixed truncation order, we prove that the regularized minimizers converge to the finite-dimensional minimum-norm solution as the noise level and the regularization parameter vanish, under a suitable choice of the regularization parameter. Some two-dimensional numerical examples are presented to illustrate the performance of the proposed method.

Alain Bensoussan, Minh-Binh Tran, Bangjie Wang

Although there is a substantial body of literature on control and optimization problems for parabolic and hyperbolic systems, the specific problem of controlling and optimizing the coefficients of the associated operators within such systems has not yet been thoroughly explored. In this work, we aim to initiate a line of research in control theory focused on optimizing and controlling the coefficients of these operators-a problem that naturally arises in the context of neural networks and supervised learning. In supervised learning, the primary objective is to transport initial data toward target data through the layers of a neural network. We propose a novel perspective: neural networks can be interpreted as partial differential equations (PDEs). From this viewpoint, the control problem traditionally studied in the context of ordinary differential equations (ODEs) is reformulated as a control problem for PDEs, specifically targeting the optimization and control of coefficients in parabolic and hyperbolic operators. To the best of our knowledge, this specific problem has not yet been systematically addressed in the control theory of PDEs. To this end, we propose a dual system formulation for the control and optimization problem associated with parabolic PDEs, laying the groundwork for the development of efficient numerical schemes in future research. We also provide a theoretical proof showing that the control and optimization problem for parabolic PDEs admits minimizers. Finally, we investigate the control problem associated with hyperbolic PDEs and prove the existence of solutions for a corresponding approximated control problem.

Alain Bensoussan, Yiqun Li, Dinh Phan Cao Nguyen et al.

We survey in this article the connections between Machine Learning and Control Theory. Control Theory provide useful concepts and tools for Machine Learning. Conversely Machine Learning can be used to solve large control problems. In the first part of the paper, we develop the connections between reinforcement learning and Markov Decision Processes, which are discrete time control problems. In the second part, we review the concept of supervised learning and the relation with static optimization. Deep learning which extends supervised learning, can be viewed as a control problem. In the third part, we present the links between stochastic gradient descent and mean-field theory. Conversely, in the fourth and fifth parts, we review machine learning approaches to stochastic control problems, and focus on the deterministic case, to explain, more easily, the numerical algorithms.

Sriramkrishnan Muralikrishnan, Minh-Binh Tran, Tan Bui-Thanh

We present a scalable iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of linear partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations, and hence inheriting advances from both sides. In particular, the method can be viewed as a Gauss-Seidel approach that requires only independent element-by-element and face-by-face local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. Unlike conventional Gauss-Seidel schemes which are purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical flux, does not depend on the ordering of unknowns. We rigorously show the convergence of the proposed method for the transport equation, the linearized shallow water equation and the convection-diffusion equation. For the transport equation, the method is convergent regardless of mesh size $h$ and solution order $p$, and furthermore the convergence rate is independent of the solution order. For the linearized shallow water and the convection-diffusion equations we show that the convergence is conditional on both $h$ and $p$. Extensive steady and time-dependent numerical results for the 2D and 3D transport equations, the linearized shallow water equation, and the convection-diffusion equation are presented to verify the theoretical findings.

Erich L Foster, Jérôme Lohéac, Minh-Binh Tran

In this paper we introduce a numerical scheme which preserves the long time behavior of solutions to the Kolmogorov equation. The method presented is based on a self-similar change of variables technique to transform the Kolmogorov equation into a new form, such that the problem of designing structure preserving schemes, for the original equation, amounts to building a standard scheme for the transformed equation. We also present an analysis for the operator splitting technique for the self-similar method and numerical results for the described scheme.