arXiv subjects

Haomin Zhou

Publications and source records attributed to Haomin Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Parameterized Methods for Game Dynamics

We introduce a parameterized computational framework for the evolution of strategic behavior in continuous games. We consider the collective dynamics of players through a time-dependent probability density over the strategy space, representing the likelihood of each strategy being chosen at any given time. Instead of directly solving the high-dimensional Fokker-Planck equation that governs this evolution, we represent the probability density as the pushforward of a reference distribution with parameterized pushforward maps and consider the evolution of the parameterized equations. The motivation for this work comes from a limitation of the parameterized Wasserstein gradient flow (PWGF) framework when it is applied to game dynamics. PWGF provides a parameterized approach for evolution equations of probability density that can be formulated as Wasserstein gradient flows. However, not all game dynamics admit such a gradient flow formulation. We generalize the parameterized pushforward map framework to the non-gradient flows and apply it to the game dynamics with provable error bound in Wasserstein metric. Numerical experiments with several non-gradient systems in economics are provided to demonstrate the effectiveness of this new framework.

math.NA↗

arXiv2026

Which Review Aspect Has a Greater Impact on the Duration of Open Peer Review in Multiple Rounds? -- Evidence from Nature Communications

Purpose: Peer review is essential to scientific publishing, but increasing submission volumes have placed growing pressure on reviewers and editors. This study examines the relationship between sentiment toward specific review aspects and peer review duration. It also investigates how this relationship varies across disciplines and review rounds, with the aim of supporting targeted manuscript revision and improving review efficiency. Design/methodology/approach: We adopt a two-stage approach. First, fine-grained aspects are extracted from peer review reports, and a sentiment classification model is used to determine the sentiment associated with each aspect. Second, correlations between aspect-level sentiment and peer review duration are analyzed. Sentiment scores are also calculated for different review rounds to determine whether these relationships change over successive rounds. Findings: Review sentiment has a weak but statistically significant negative correlation with peer review duration, indicating that more positive reviews tend to be associated with shorter review periods. Aspects concerning Evaluation and Results and Impact and Research Value show relatively stronger correlations with review duration. The relationships between aspect-level sentiment and review duration also differ significantly across review rounds. Originality/value: This study connects the textual content of peer review reports with the temporal characteristics of the review process. By identifying review aspects that are more closely associated with review duration, it provides evidence that may help authors prioritize revisions and assist reviewers and editors in improving review efficiency. The findings contribute to reducing the burden of peer review and accelerating scholarly communication and knowledge dissemination.

cs.CL↗

arXiv2026

Newton's method for optimal transport problem on graphs

In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis.

math.NA↗

arXiv2026

Discrete Mean Field Games on Finite Graphs as Initial Value Optimization

In this paper, we propose an initial value fomulation of the discrete mean field games on finite graphs (Graph MFG), and design a neural network based approach to solve it. Graph MFG describes infinite, non-cooperative and interactive homogeneous agents move on node states through the edges to optimize their own goals. Nash Equilibrium of the Graph MFG is characterized by a coupled ordinary differential equations (ODE) system, including the discrete forward continuity equation and the discrete backward Hamilton-Jacobi equation. In this paper, we mainly focus on the potential mean field games (Potential MFG) on finite graphs, which has an infinite-dimensional constrained optimization structure. We reformulate Potential MFG as an initial value finite-dimentional optimization problem with dynamics constrains, names Graph MFG-IV. Specifically, the initial condition of the Hamilton-Jacobi equation is regarded as the unique variable, constrained by the coupled Hamilton-Jacobi and continuity equation system as the ODE integrator. This formulation is a reduced-order model, which avoids time-discretization of the infinite-dimensional path and has a much smaller searching space than the general path-wise problem setting. We design a neural network-based approach to solve the Graph MFG-IV problem.

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arXiv2026

Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space

We compute the ground state $u$ of the Gross--Pitaevskii equation (GPE) via Wasserstein gradient descent in diffeomorphism space. We represent the density $\rho=u^2$ as the push-forward of a fixed reference measure through a parameterized transport map $T_\theta$, realized by a boundary-preserving Neural ODE. The Wasserstein gradient flow on probability densities then lifts to natural gradient descent in the finite-dimensional parameter space, with metric tensor given by the pullback of the Wasserstein metric. The method is entirely mesh-free and preserves the unit-mass constraint without normalization. We present numerical experiments in dimensions $d=1$ to $4$, with separable and non-separable potentials, using a product transport map in the separable case and a full map with coupled components in the non-separable one. The tests suggest that the output of the parameterized Wasserstein gradient flow (PWGF) can be used as an effective warm start for conventional solvers such as the $H^1$ Sobolev gradient flow.

math.NA↗

arXiv2026

Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications

In the current work, we propose a generalization of angles and orthogonality from $L^2$ to generic Banach spaces, starting from a $L^p$ version of the Pythagorean theorem, $p\in [1,\infty)$. The starting point is conservation of energy measured in $L^1$ norm, as it occurs when considering the intrinsic mode functions decomposition in signal processing. This conservation of energy measure in $L^1$ norm is exactly the $L^1$ Pythagorean theorem. Besides the theoretical analysis, we apply the new notions in the context of preconditioning for structured large linear systems, by obtaining new classes of preconditioners. The present work contains numerical experiments and various remarks on the possible use of the given framework.

math.GM↗

arXiv2026

Multi-agent path-planning in a moving medium via Wasserstein Hamiltonian Flow

We present a finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium. The model is derived using the agent-based formulation of the Wasserstein Hamiltonian flows that transport between probability distributions while optimizing a running cost. The objective is the mismatch between their final positions and the target distribution. The constraints are a system of Hamiltonian equations that provide the trajectories of the agents. The free variables on which the optimization is defined form a finite vector of the initial velocities for the agents. The model is solved numerically by the L-BFGS method in conjunction with a shooting strategy. Several simulation examples, including a time-dependent moving medium, are presented to illustrate the performance of the model.

math.OC↗

arXiv2025

Deep Tangent Bundle (DTB) method: a Deep Neural Network approach to compute solutions of PDES

We develop a numerical framework, the Deep Tangent Bundle (DTB) method, that is suitable for computing solutions of evolutionary partial differential equations (PDEs) in high dimensions. The main idea is to use the tangent bundle of an adaptively updated deep neural network (DNN) to approximate the vector field in the spatial variables while applying the traditional schemes for time discretization. The DTB method takes advantage of the expression power of DNNs and the simplicity of the tangent bundle approximation. It does not involve nonconvex optimization. Several numerical examples demonstrate that the DTB is simple, flexible, and efficient for various PDEs of higher dimensions.

math.NA↗

arXiv2025

Identification of Differential Equations by Dynamics-Guided Weighted Weak Form with Voting

In the identification of differential equations from data, significant progresses have been made with the weak/integral formulation. In this paper, we explore the direction of finding more efficient and robust test functions adaptively given the observed data. While this is a difficult task, we propose weighting a collection of localized test functions for better identification of differential equations from a single trajectory of noisy observations on the differential equation. We find that using high dynamic regions is effective in finding the equation as well as the coefficients, and propose a dynamics indicator per differential term and weight the weak form accordingly. For stable identification against noise, we further introduce a voting strategy to identify the active features from an ensemble of recovered results by selecting the features that frequently occur in different weighting of test functions. Systematic numerical experiments are provided to demonstrate the robustness of our method.

math.NA↗

arXiv2025

PDPO: Parametric Density Path Optimization

We introduce Parametric Density Path Optimization (PDPO), a novel method for computing action-minimizing paths between probability densities. The core idea is to represent the target probability path as the pushforward of a reference density through a parametric map, transforming the original infinite-dimensional optimization over densities to a finite-dimensional one over the parameters of the map. We derive a static formulation of the dynamic problem of action minimization and propose cubic spline interpolation of the path in parameter space to solve the static problem. Theoretically, we establish an error bound of the action under proper assumptions on the regularity of the parameter path. Empirically, we find that using 3-5 control points of the spline interpolation suffices to accurately resolve both multimodal and high-dimensional problems. We demonstrate that PDPO can flexibly accommodate a wide range of potential terms, including those modeling obstacles, mean-field interactions, stochastic control, and higher-order dynamics. Our method outperforms existing state-of-the-art approaches in benchmark tasks, demonstrating superior computational efficiency and solution quality. The source code will be publically available after the revision process.

math.OC↗

arXiv2025

A parameterized Wasserstein Hamiltonian flow approach for solving the Schr\"odinger equation

In this paper, we propose a new method to compute the solution of time-dependent Schr\"odinger equation (TDSE). Using push-forward maps and Wasserstein Hamiltonian flow, we reformulate the TDSE as a Hamiltonian system in terms of push-forward maps. The new formulation can be viewed as a generative model in the Wasserstein space, which is a manifold of probability density functions. Then we parameterize the push-forward maps by reduce-order models such as neural networks. This induces a new metric in the parameter space by pulling back the Wasserstein metric on density manifold, which further results in a system of ordinary differential equations (ODEs) for the parameters of the reduce-order model. Leveraging the computational techniques from deep learning, such as Neural ODE, we design an algorithm to solve the TDSE in the parameterized push-forward map space, which provides an alternative approach with the potential to scale up to high-dimensional problems. Several numerical examples are presented to demonstrate the performance of this algorithm.

math.NA↗

arXiv2025

Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potential

The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are well matched, as their alignment is key to effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a model that combines sine-type activations with the multi-component and multi-layer structure of MMNNs. In an FMMNN, each component is represented as a trainable linear combination of fixed random sine-type basis functions, while multi-layer composition generates more complex and adaptive high-frequency features. We establish that FMMNNs retain exponential expressive power for function approximation even under a low-rank architectural structure. We also analyze the optimization landscape of FMMNNs and find it to be substantially more favorable than that of standard fully connected neural networks, especially for high-frequency targets. In addition, we propose a scaled random initialization method for the first-layer weights in FMMNNs, which accelerates training and improves final performance when sufficient samples are available. Extensive numerical experiments support our theoretical insights, showing that FMMNNs achieve strong accuracy and favorable convergence behavior on oscillatory function-approximation benchmarks.

cs.LG↗

arXiv2024

A novel algorithm for the decomposition of non-stationary multidimensional and multivariate signals

The decomposition of a signal is a fundamental tool in many fields of research, including signal processing, geophysics, astrophysics, engineering, medicine, and many more. By breaking down complex signals into simpler oscillatory components we can enhance the understanding and processing of the data, unveiling hidden information contained in them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in handling mono-dimensional stationary signals struggle with non-stationary data sets and they require, this is the case of the wavelet, the selection of predefined basis functions. In contrast, the Empirical Mode Decomposition (EMD) method and its variants, such as Iterative Filtering (IF), have emerged as effective nonlinear approaches, adapting to signals without any need for a priori assumptions. To accelerate these methods, the Fast Iterative Filtering (FIF) algorithm was developed, and further extensions, such as Multivariate FIF (MvFIF) and Multidimensional FIF (FIF2), have been proposed to handle higher-dimensional data. In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique, an innovative method that extends FIF to handle data that vary simultaneously in space and time. This new algorithm is capable of extracting Intrinsic Mode Functions (IMFs) from complex signals that vary in both space and time, overcoming limitations found in prior methods. The potentiality of the proposed method is demonstrated through applications to artificial and real-life signals, highlighting its versatility and effectiveness in decomposing multidimensional and multivariate nonstationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis across many scientific and engineering disciplines.

math.NA↗

arXiv2024

Structured and Balanced Multi-Component and Multi-Layer Neural Networks

In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to accurately and efficiently approximate functions with complex features, in terms of both degrees of freedom and computational cost. The main idea is inspired by a multi-component approach, in which each component can be effectively approximated by a single-layer network, combined with a multi-layer decomposition strategy to capture the complexity of the target function. Although MMNNs can be viewed as a simple modification of fully connected neural networks (FCNNs) or multi-layer perceptrons (MLPs) by introducing balanced multi-component structures, they achieve a significant reduction in training parameters, a much more efficient training process, and improved accuracy compared to FCNNs or MLPs. Extensive numerical experiments demonstrate the effectiveness of MMNNs in approximating highly oscillatory functions and their ability to automatically adapt to localized features.

cs.LG↗

arXiv2024

Parameterized Wasserstein Gradient Flow

We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), particularly suitable for high-dimensional cases. Our approach is to use general reduced-order models, like deep neural networks, to parameterize the push-forward maps such that they can push a simple reference density to the one solving the given WGF. The new dynamical system is called parameterized WGF (PWGF), and it is defined on the finite-dimensional parameter space equipped with a pullback Wasserstein metric. Our numerical scheme can approximate the solutions of WGFs for general energy functionals effectively, without requiring spatial discretization or nonconvex optimization procedures, thus avoiding some limitations of classical numerical methods and more recent deep-learning-based approaches. A comprehensive analysis of the approximation errors measured by Wasserstein distance is also provided in this work. Numerical experiments show promising computational efficiency and verified accuracy on various WGF examples using our approach.

math.NA↗

arXiv2024

A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling

We propose a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on $L^1$ residual of the proposed scheme based on the coupling density. Furthermore, the proposed scheme can be used to describe the behaviors of Hamilton--Jacobi PDEs beyond the singularity formations on the support of coupling density. Several numerical examples with different Hamiltonians are provided to support our findings.

math.NA↗

arXiv2023

Neural Control of Parametric Solutions for High-dimensional Evolution PDEs

We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.

math.NA↗

arXiv2023

IRCNN$^{+}$: An Enhanced Iterative Residual Convolutional Neural Network for Non-stationary Signal Decomposition

Time-frequency analysis is an important and challenging task in many applications. Fourier and wavelet analysis are two classic methods that have achieved remarkable success in many fields. However, they also exhibit limitations when applied to nonlinear and non-stationary signals. To address this challenge, a series of nonlinear and adaptive methods, pioneered by the empirical mode decomposition method, have been proposed. The goal of these methods is to decompose a non-stationary signal into quasi-stationary components that enhance the clarity of features during time-frequency analysis. Recently, inspired by deep learning, we proposed a novel method called iterative residual convolutional neural network (IRCNN). IRCNN not only achieves more stable decomposition than existing methods but also handles batch processing of large-scale signals with low computational cost. Moreover, deep learning provides a unique perspective for non-stationary signal decomposition. In this study, we aim to further improve IRCNN with the help of several nimble techniques from deep learning and optimization to ameliorate the method and overcome some of the limitations of this technique.

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