Minh-Nhat Phung

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.

MaryLena Bleile, Minh-Nhat Phung, Minh-Binh Tran

Sequential decision-making systems routinely operate with missing or incomplete data. Classical reinforcement learning theory, which is commonly used to solve sequential decision problems, assumes Markovian observability, which may not hold under partial observability. Causal inference paradigms formalise ignorability of missingness. We show these views can be unified and generalized in order to guarantee Q-learning convergence even when the Markov property fails. To do so, we introduce the concept of relative ignorability. Relative ignorability is a graphical-causal criterion which refines the requirements for accurate decision-making based on incomplete data. Theoretical results and simulations both reveal that non-Markovian stochastic processes whose missingness is relatively ignorable with respect to causal estimands can still be optimized using standard Reinforcement Learning algorithms. These results expand the theoretical foundations of safe, data-efficient AI to real-world environments where complete information is unattainable.

Minh-Nhat Phung, Minh-Binh Tran

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.