arXiv subjects

Ruimeng Hu

Publications and source records attributed to Ruimeng Hu.

At least 37 records · Page 2Linked to original sources

arXiv2026

Convergence Analysis of PINNs for Fractional Diffusion Equations in Bounded Domains

We establish the convergence of physics-informed neural networks (PINNs) for time-dependent fractional diffusion equations posed on bounded domains. The presence of fractional Laplacian operators introduces nonlocal behavior and regularity constraints, and standard neural network approximations do not naturally enforce the associated spectral boundary conditions. To address this challenge, we introduce a spectrally-defined mollification strategy that preserves the structure of the nonlocal operator while ensuring boundary compatibility. This enables the derivation of rigorous energy estimates in Sobolev spaces. Our results rely on analytical tools from PDE theory, highlighting the compatibility of PINN approximations with classical energy estimates for nonlocal equations. We prove convergence of the PINN approximation in any space-time Sobolev norm $H^k$ (with $k \in \N)$. The analysis highlights the role of mollified residuals in enabling theoretical guarantees for neural-network-based solvers of nonlocal PDEs.

math.NA↗

arXiv2025

Error estimates of physics-informed neural networks for approximating Boltzmann equation

Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks.

math.NA↗

arXiv2025

Finite-Agent Stochastic Differential Games on Large Graphs: I. The Linear-Quadratic Case

In this paper, we study finite-agent linear-quadratic games on graphs. Specifically, we propose a comprehensive framework that extends the existing literature by incorporating heterogeneous and interpretable player interactions. Compared to previous works, our model offers a more realistic depiction of strategic decision-making processes. For general graphs, we establish the convergence of fictitious play, a widely-used iterative solution method for determining the Nash equilibrium of our proposed game model. Notably, under appropriate conditions, this convergence holds true irrespective of the number of players involved. For vertex-transitive graphs, we develop a semi-explicit characterization of the Nash equilibrium. Through rigorous analysis, we demonstrate the well-posedness of this characterization under certain conditions. We present numerical experiments that validate our theoretical results and provide insights into the intricate relationship between various game dynamics and the underlying graph structure.

math.OC↗

arXiv2025

Deep Signature Approach for McKean-Vlasov FBSDEs in a Random Environment

Mean-field games with common noise provide a powerful framework for modeling the collective behavior of large populations subject to shared randomness, such as systemic risk in finance or environmental shocks in economics. These problems can be reformulated as McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) in a random environment, where the coefficients depend on the conditional law of the state given the common noise. Existing numerical methods, however, are largely limited to cases where interactions depend only on expectations or low-order moments, and therefore cannot address the general setting of full distributional dependence. In this work, we introduce a deep learning-based algorithm for solving MV-FBSDEs with common noise and general mean-field interactions. Building on fictitious play, our method iteratively solves conditional FBSDEs with fixed distributions, where the conditional law is efficiently represented using signatures, and then updates the distribution through supervised learning. Deep neural networks are employed both to solve the conditional FBSDEs and to approximate the distribution-dependent coefficients, enabling scalability to high-dimensional problems. Under suitable assumptions, we establish convergence in terms of the fictitious play iterations, with error controlled by the supervised learning step. Numerical experiments, including a distribution-dependent mean-field game with common noise, demonstrate the effectiveness of the proposed approach.

math.OC↗

arXiv2025

Learning Mean-Field Games through Mean-Field Actor-Critic Flow

We propose the Mean-Field Actor-Critic (MFAC) flow, a continuous-time learning dynamics for solving mean-field games (MFGs), combining techniques from reinforcement learning and optimal transport. The MFAC framework jointly evolves the control (actor), value function (critic), and distribution components through coupled gradient-based updates governed by partial differential equations (PDEs). A central innovation is the Optimal Transport Geodesic Picard (OTGP) flow, which drives the distribution toward equilibrium along Wasserstein-2 geodesics. We conduct a rigorous convergence analysis using Lyapunov functionals and establish global exponential convergence of the MFAC flow under a suitable timescale. Our results highlight the algorithmic interplay among actor, critic, and distribution components. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the MFAC framework in computing MFG equilibria.

math.OC↗

arXiv2025

Integrating Sequential Hypothesis Testing into Adversarial Games: A Sun Zi-Inspired Framework

This paper investigates the interplay between sequential hypothesis testing (SHT) and adversarial decision-making in partially observable games, focusing on the deceptive strategies of red and blue teams. Inspired by Sun Zi's The Art of War and its emphasis on deception, we develop a novel framework to both deceive adversaries and counter their deceptive tactics. We model this interaction as a Stackelberg game where the blue team, as the follower, optimizes its controls to achieve its goals while misleading the red team into forming incorrect beliefs on its intentions. The red team, as the leader, strategically constructs and instills false beliefs through the blue team's envisioned SHT to manipulate the blue team's behavior and reveal its true objectives. The blue team's optimization problem balances the fulfillment of its primary objectives and the level of misdirection, while the red team coaxes the blue team into behaving consistently with its actual intentions. We derive a semi-explicit solution for the blue team's control problem within a linear-quadratic framework, and illustrate how the red team leverages leaked information from the blue team to counteract deception. Numerical experiments validate the model, showcasing the effectiveness of deception-driven strategies in adversarial systems. These findings integrate ancient strategic insights with modern control and game theory, providing a foundation for further exploration in adversarial decision-making, such as cybersecurity, autonomous systems, and financial markets.

math.OC↗

arXiv2025

Estimation of Anisotropic Viscosities for the Stochastic Primitive Equations

The viscosity parameters play a fundamental role in applications involving stochastic primitive equations (SPE), such as accurate weather predictions, climate modeling, and ocean current simulations. In this paper, we develop several novel estimators for the anisotropic viscosities in the SPE, using a finite number of Fourier modes of a single sample path observed within a finite time interval. The focus is on analyzing the consistency and asymptotic normality of these estimators. We consider a torus domain and treat strong, pathwise solutions in the presence of additive white noise (in time). Notably, the analysis for estimating horizontal and vertical viscosities differs due to the unique structure of the SPE and the fact that both parameters of interest are adjacent to the highest-order derivative. To the best of our knowledge, this is the first work addressing the estimation of anisotropic viscosities, with the potential applicability of the developed methodology to other models.

math.PR↗

arXiv2025

Multi-Agent Relative Investment Games in a Jump Diffusion Market with Deep Reinforcement Learning Algorithm

This paper focuses on multi-agent stochastic differential games for jump-diffusion systems. On one hand, we study the multi-agent game for optimal investment in a jump-diffusion market. We derive constant Nash equilibria and provide sufficient conditions for their existence and uniqueness for exponential, power, and logarithmic utilities, respectively. On the other hand, we introduce a computational framework based on the actor-critic method in deep reinforcement learning to solve the stochastic control problem with jumps. We extend this algorithm to address the multi-agent game with jumps and utilize parallel computing to enhance computational efficiency. We present numerical examples of the Merton problem with jumps, linear quadratic regulators, and the optimal investment game under various settings to demonstrate the accuracy, efficiency, and robustness of the proposed method. In particular, neural network solutions numerically converge to the derived constant Nash equilibrium for the multi-agent game.

math.OC↗

arXiv2025

Deep Reinforcement Learning for Infinite Horizon Mean Field Problems in Continuous Spaces

We present the development and analysis of a reinforcement learning (RL) algorithm designed to solve continuous-space mean field game (MFG) and mean field control (MFC) problems in a unified manner. The proposed approach pairs the actor-critic (AC) paradigm with a representation of the mean field distribution via a parameterized score function, which can be efficiently updated in an online fashion, and uses Langevin dynamics to obtain samples from the resulting distribution. The AC agent and the score function are updated iteratively to converge, either to the MFG equilibrium or the MFC optimum for a given mean field problem, depending on the choice of learning rates. A straightforward modification of the algorithm allows us to solve mixed mean field control games (MFCGs). The performance of our algorithm is evaluated using linear-quadratic benchmarks in the asymptotic infinite horizon framework.

math.OC↗

arXiv2025

On the local well-posedness of fractionally dissipated primitive equations with transport noise

We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by $ (-Δ)^{s/2} $ with $ s \in (1,2)$ and $s = 1$, respectively. For $σ>3$, we establish the local existence of unique pathwise solutions in Sobolev space $H^σ$. This result applies to arbitrary initial data in the subcritical case ($s \in(1,2)$), and to small initial data in the critical case ($s=1$). The analysis is particularly challenging due to the loss of horizontal derivatives in the nonlinear terms and the lack of full dissipation. To address these challenges, we develop novel commutator estimates involving the hydrostatic Leray projection.

math.AP↗

arXiv2024

A PDE Perspective on Approximating Nonlocal Periodic Operators with Applications on Neural Networks for Critical SQG Equations

Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.

math.NA↗

arXiv2024

Regularization by noise for the inviscid primitive equations

The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only locally in time and exhibit finite-time blowup. This work demonstrates that introducing suitable random noise can restore the local well-posedness and prevent finite-time blowups. Specifically, random diffusion addresses the ill-posedness in certain Gevrey classes, allowing us to establish the local well-posedness almost surely and the global existence of solutions with high probability. In the case of random damping (linear multiplicative noise), the noise prevents analytic solutions from forming singularities in finite time, resulting in globally existing solutions with high probability.

math.AP↗

arXiv2024

On the Three-dimensional Nernst-Planck-Boussinesq System

In this paper, we analyze a three-dimensional Nernst-Planck-Boussinesq (NPB) system that describes ionic electrodiffusion in an incompressible viscous fluid. This new model incorporates variational temperature and is forced by buoyancy force stemming from temperature and salinity fluctuations, enhancing its generality and realism. The electromigration term in the NPB system displays a complex nonlinear structure influenced by the reciprocal of the temperature that distinguishes its mathematical aspects from other electrodiffusion models studied in the literature. We address the global existence of weak solutions to the NPB system on the three-dimensional torus for large initial data. In addition, we study the long-time dynamics of these weak solutions and the associated relative entropies and establish their exponential decay in time to steady states.

math.AP↗

arXiv2024

Recent Developments in Machine Learning Methods for Stochastic Control and Games

Stochastic optimal control and games have a wide range of applications, from finance and economics to social sciences, robotics, and energy management. Many real-world applications involve complex models that have driven the development of sophisticated numerical methods. Recently, computational methods based on machine learning have been developed for solving stochastic control problems and games. In this review, we focus on deep learning methods that have unlocked the possibility of solving such problems, even in high dimensions or when the structure is very complex, beyond what traditional numerical methods can achieve. We consider mostly the continuous time and continuous space setting. Many of the new approaches build on recent neural-network-based methods for solving high-dimensional partial differential equations or backward stochastic differential equations, or on model-free reinforcement learning for Markov decision processes that have led to breakthrough results. This paper provides an introduction to these methods and summarizes the state-of-the-art works at the crossroad of machine learning and stochastic control and games.

math.OC↗

arXiv2023

On the Long-time Dynamics and Ergodicity of the Stochastic Nernst-Planck-Navier-Stokes System

We consider an electrodiffusion model that describes the intricate interplay of multiple ionic species with a two-dimensional, incompressible, viscous fluid subjected to stochastic additive noise. This system involves nonlocal nonlinear drift-diffusion Nernst-Planck equations for ionic species and stochastic Navier-Stokes equations for fluid motion under the influence of electric and time-independent forces. Under the selective boundary conditions imposed on the concentrations, we establish the existence and uniqueness of global pathwise solutions to this system on smooth bounded domains. Our study also investigates long-time ionic concentration dynamics and explores Feller properties of the associated Markovian semigroup. In the context of equal diffusive species and under appropriate conditions, we demonstrate the existence of invariant ergodic measures supported on $H^2$. We then enhance the ergodicity results on periodic tori and obtain smooth invariant measures under a constraint on the initial spatial averages of the concentrations. The uniqueness of the invariant measures on periodic boxes and smooth bounded domains is further established when the noise forces sufficient modes, and the diffusivities of the species are large. Finally, in the case of two ionic species with equal diffusivities and valences of $1$ and $-1$, we study the rate of convergence of the Markov transition kernels to the invariant measure and obtain unconditional, unique exponential ergodicity for the model.

math.AP↗

arXiv2023

A Deep Learning Analysis of Climate Change, Innovation, and Uncertainty

We study the implications of model uncertainty in a climate-economics framework with three types of capital: "dirty" capital that produces carbon emissions when used for production, "clean" capital that generates no emissions but is initially less productive than dirty capital, and knowledge capital that increases with R\&D investment and leads to technological innovation in green sector productivity. To solve our high-dimensional, non-linear model framework we implement a neural-network-based global solution method. We show there are first-order impacts of model uncertainty on optimal decisions and social valuations in our integrated climate-economic-innovation framework. Accounting for interconnected uncertainty over climate dynamics, economic damages from climate change, and the arrival of a green technological change leads to substantial adjustments to investment in the different capital types in anticipation of technological change and the revelation of climate damage severity.

econ.GN↗

arXiv2023

Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs' model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration's FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward process.

math.OC↗

arXiv2023

A Class of Dimension-free Metrics for the Convergence of Empirical Measures

This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics ({\it e.g.}, the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of $n$-particle system to the solution to McKean-Vlasov stochastic differential equation; 3. The construction of an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.

math.PR↗