arXiv subjects

Minh-Binh Tran

Publications and source records attributed to Minh-Binh Tran.

At least 19 recordsLinked to original sources

arXiv2026

Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control

We establish a finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems. Geometric supervision is used only during training: simulator state, proprioception, or state estimates with independently validated metric and directional error bounds supply observable-state distances and tangent directions, while deployment remains observation- and action-conditioned. We introduce an encoder-only local--global metric hinge that enforces directional resolution and separated-state discrimination. Under regular observable-factor, coverage, finite-capacity approximation, and uniform $C^{1,1}$ hypotheses, a computable one-sided regularization regime has a strong selection property: with high probability, every approximate empirical minimizer is simultaneously pointwise co-Lipschitz and uniformly approximately semiconjugate to the controlled dynamics. Approximation, sampling, and optimization errors remain explicit and separate. Norm-constrained tensor-product B-spline classes constructively realize the approximation hypotheses, and the interpolation exponent converting mean residual control into a uniform bound is sharp. A modular deterministic corollary transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer guarantees, while a validated finite-net result enables sharper model-specific certification. Controlled experiments isolate collapse and folding, quantify the analytic certificate's reserve, and demonstrate the control benefit of restored metric resolution. The principal contribution is a complete finite-sample implication from approximate empirical optimization to metric faithfulness, uniform controlled dynamics, and reliable planning for the same learned model.

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arXiv2026

Spectral Algorithms for 3-Wave Kinetic and $C_{12}$ Quantum Boltzmann Equations with General Resonance Manifolds in $\mathbb{R}^d$

Following recent developments in numerical schemes for 3-wave kinetic equations [2, 7, 42, 44, 43], we develop spectral algorithms for multidimensional 3-wave kinetic equations and $C_{12}$ quantum Boltzmann equations with general polynomial dispersion relations. The principal numerical difficulty arises from the resonance constraint, supported on a nonlinear manifold in wave-vector space. We approximate the Dirac distribution by a truncated Fourier representation and derive two spectral discretizations of the collision operator. The first is a direct spectral method with complexity $\mathcal{O}\big(L(2N)^{3d}\big)$, while the second exploits multidimensional FFTs to reduce the complexity to $\mathcal{O}\big(L(2N)^{2d}\log(2N)\big)$. Numerical tests show excellent agreement between the two methods, with the fast algorithm providing substantial computational savings. To suppress unresolved high-frequency modes, we combine the classical $2/3$-rule with exponential spectral filtering. Simulations in two and three dimensions capture the gain--loss dynamics of the $C_{12}$ quantum Boltzmann equation for both rapidly and algebraically decaying initial data. For the 3-wave kinetic equation, the computations exhibit strong oscillations and rapid spectral broadening, providing numerical evidence of an apparent energy cascade toward high frequencies. The results also show that the dispersion relation and spatial dimension strongly influence the transient resonant dynamics.

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arXiv2026

Entropy Structures and Long-Time Relaxation for 3-Wave Kinetic Equations

We establish a new class of entropy structures for $3$-wave kinetic equations with a broad family of interaction weights. Unlike the classical entropies arising from detailed balance, these estimates are generated by a one-sided algebraic balance condition encoded in the interaction weights. To the best of our knowledge, this family of entropy estimates has not previously appeared in the physical literature on wave turbulence. These estimates form the central a priori mechanism of the paper and are the key ingredient in the construction of global weak $L^1_{\mathrm{loc}}$ solutions. We also prove a long-time rigidity result, showing that the solutions obtained by this entropy compactness method relax locally to the zero equilibrium as $t\to\infty$.

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arXiv2026

Global time-analytic strong solutions for a class of 3-wave kinetic equations

We study a class of 3-wave kinetic equations arising in wave turbulence theory, with regularized kernels. For radial, nonnegative initial data, we construct an exact global-in-time strong solution which remains nonnegative and is analytic with respect to time. The proof combines a careful analysis of the resonant interaction surfaces with a time power-series construction and a continuation argument based on the conservation of the energy moment.

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arXiv2026

A Structure-Preserving Neural-Spectral Method for Reconstructing Controls of Wave Equations

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.

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arXiv2026

Inverse initial data reconstruction for a memory convection-diffusion equation via Legendre spatial reduction and Tikhonov regularization

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.

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arXiv2026

A Microlocal Open-Boundary Method for Residual-Based Wave Solvers on Unbounded Domains

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.

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arXiv2026

Computational Control of Nonlinear Partial Differential Equations Using Machine Learning

The numerical reconstruction of controls for nonlinear partial differential equations (PDEs) remains a challenging and relatively underdeveloped problem, despite the extensive literature on controllability theory. In this work, we introduce an operator-decomposed physics-informed neural network framework, called WeightedPINN, for approximating controls in nonlinear PDE settings. The method is designed for both internal and bilinear control problems and incorporates the governing equation, boundary and initial conditions, and terminal control constraints directly into the training objective. The main feature of WeightedPINN is that the different components of the controlled PDE residual are weighted separately. In particular, the time derivative, directional diffusion terms, nonlinear response, and control term are assigned independent adaptive space--time weights, and the same weighted formulation is applied to the boundary, initial, and terminal constraints. This produces a control-aware residual metric that is more sensitive to operator-level imbalance and to the mechanism through which the unknown control enters the equation. We provide a convergence analysis for the proposed method and present numerical experiments for semilinear heat and wave equations with internal and bilinear controls. The high-dimensional experiments demonstrate improved residual-based testing errors compared with the standard PINN baseline, while lower-dimensional manufactured-solution benchmarks show improved direct reconstruction errors for both the state and the control against several adaptive and control-oriented PINN methods. The results suggest that WeightedPINN is particularly effective in regimes where componentwise residual imbalance, anisotropy, variable coefficients, or control-identification sensitivity play a significant role.

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arXiv2026

Control, Optimal Transport and Neural Differential Equations in Supervised Learning

We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal transport (UOT) in the continuum using Neural ODEs. By generalizing a discrete UOT problem with Pearson divergence, we constructively design vector fields for Neural ODEs that converge to the true UOT dynamics, thereby advancing the mathematical foundations of computational transport and machine learning. To this end, we design a numerical scheme inspired by the Sinkhorn algorithm to solve the corresponding minimization problem and rigorously prove its convergence, providing explicit error estimates. From the obtained numerical solutions, we derive vector fields defining the transport dynamics and construct the corresponding transport equation. Finally, from the numerically obtained transport equation, we construct a neural differential equation whose flow converges to the true transport dynamics in an appropriate limiting regime.

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arXiv2026

Control and optimization for Neural Partial Differential Equations in Supervised Learning

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.

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arXiv2026

On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas

This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime $t\sim λ^{-2}$ where $λ\ll 1$ is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.

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arXiv2026

A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation

We study an inverse initial-density problem for a nonlinear diffusive coagulation--fragmentation equation with known coagulation and fragmentation kernels. The objective is to recover the unknown initial particle-size distribution on a finite interval from time-dependent boundary observations of the solution and its size derivative. To solve this inverse problem, we develop a globally convergent numerical method based on a Legendre--exponential time reduction and a Carleman--Picard iteration. The time reduction transforms the original problem into a nonlinear coupled system for the spatial mode coefficients, while the Carleman weight and the corresponding Carleman estimate guaranty the global convergence of the Picard iteration without requiring a good initial guess. We prove the convergence of the proposed method and obtain a complete reconstruction procedure for the initial density. Numerical experiments with noisy boundary data demonstrate that the method yields accurate and stable reconstructions for several representative test profiles.

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arXiv2026

Operator Splitting, Policy Iteration, and Machine Learning for Stochastic Optimal Control

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, with $h$ the splitting step, the $L^\infty$ error is bounded between $\mathcal{O}(h)$ and $\mathcal{O}(h^{1/5})$ for Lipschitz data, improving to $\mathcal{O}(h^{1/3})$ for semiconcave data. In the periodic setting, we also obtain an $L^1$ error of order $\mathcal{O}(h^{1/2})$. 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.

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arXiv2026

Analysis of a numerical scheme for 3-wave kinetic equations

Several numerical schemes for 3-wave kinetic equations have been proposed in recent work and shown to be accurate and computationally efficient [8,33,34,35]. However, their rigorous numerical analysis remains open. This paper aims to close this gap. We establish a comprehensive well-posedness and qualitative theory for the discrete equation arising from those schemes. We prove global existence, uniqueness, and Lipschitz stability of nonnegative classical solutions in $\ell^1(\mathbb{N})$, together with uniform bounds and decay of moments. We further show exponential energy decay and a sharp creation and propagation of positivity characterized by the arithmetic structure of the initial support. Finally, we obtain the propagation and instantaneous creation of polynomial, Mittag-Leffler, and exponential moments, providing quantitative control of high energy tails. We validate the theoretical findings by numerical results.

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arXiv2025

Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth

In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.

math-ph↗

arXiv2025

Finite time energy cascade for mixed $3-$ and $4-$wave kinetic equations

In this work we study a kinetic equation whose collision operator comprises three distinct wave interaction mechanisms: one representing a 3-wave process, and two corresponding to 4-wave processes. This wave kinetic equation describes the temporal evolution of the density function of the thermal cloud of a finite temperature trapped Bose gas. We establish that, for a broad class of initial data, solutions exhibit an immediate cascade of energy towards arbitrarily large frequencies. Furthermore, for other classes of initial conditions, we demonstrate that the energy is transferred to infinity in finite time.

math-ph↗

arXiv2025

Controlling Klein-Gordon Chains and Lattices

In this work, we initiate the study of controlling nonlinear Klein-Gordon chains and lattices through their emergent collective flocking behavior. By constructing appropriate feedback control mechanisms, we demonstrate that any physically admissible flock state can be achieved in finite time, meaning the chain can be driven from arbitrary initial vibrations toward a coherent traveling-wave motion. Finally, we reveal a deep connection between the flocking problem and a minimal-time control principle formulated within the framework of nonlinear Hamilton-Jacobi equations and optimal control theory, providing a unifying view-point for wave control in discrete nonlinear media.

math.OC↗

arXiv2025

An energy cascade finite volume scheme for a mixed 3- and 4-wave kinetic equation arising from the theory of finite-temperature trapped Bose gases

Building on recent developments in numerical schemes designed to capture energy cascades for 3-wave kinetic equations~\cite{das2024numerical, walton2022deep, walton2023numerical, walton2024numerical}, we construct in this work a finite-volume algorithm for a significantly more complex wave kinetic equation whose collision operator incorporates both 3-wave and 4-wave interactions. This model arises in the context of finite-temperature Bose-Einstein condensation. We establish theoretical properties of the proposed scheme, and our numerical experiments demonstrate that it successfully captures the energy cascade behavior predicted by the equation.

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