Vertex Exchange Method for a Class of Quadratic Programming Problems

A vertex exchange method is proposed for solving the strongly convex quadratic program subject to the generalized simplex constraint. We conduct rigorous convergence analysis for the proposed algorithm and demonstrate its essential roles in solving some important classes of constrained convex optimization. To get a feasible initial point to execute the algorithm, we also present and analyze a highly efficient semismooth Newton method for computing the projection onto the generalized simplex. The excellent practical performance of the proposed algorithms is demonstrated by a set of extensive numerical experiments. Our theoretical and numerical results further motivate the potential applications of the considered model and the proposed algorithms.

FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model

Human-designed algorithms have long been fundamental in solving a variety of scientific and engineering challenges. Recently, data-driven deep learning methods have also risen to prominence, offering innovative solutions across numerous scientific fields. While traditional algorithms excel in capturing the core aspects of specific problems, they often lack the flexibility needed for varying problem conditions due to the absence of specific data. Conversely, while data-driven approaches utilize vast datasets, they frequently fall short in domain-specific knowledge. To bridge these gaps, we introduce \textbf{FMint} (Foundation Model based on Initialization), a generative pre-trained model that synergizes the precision of human-designed algorithms with the adaptability of data-driven methods. This model is specifically engineered for high-accuracy simulation of dynamical systems. Starting from initial trajectories provided by conventional methods, FMint quickly delivers highly accurate solutions. It incorporates in-context learning and has been pre-trained on a diverse corpus of 500,000 dynamical systems, showcasing exceptional generalization across a broad spectrum of real-world applications. By effectively combining algorithmic rigor with data-driven flexibility, FMint sets the stage for the next generation of scientific foundation models, tackling complex problems with both efficiency and high accuracy.

On the Stochastic (Variance-Reduced) Proximal Gradient Method for Regularized Expected Reward Optimization

We consider a regularized expected reward optimization problem in the non-oblivious setting that covers many existing problems in reinforcement learning (RL). In order to solve such an optimization problem, we apply and analyze the classical stochastic proximal gradient method. In particular, the method has shown to admit an $O(\epsilon^{-4})$ sample complexity to an $\epsilon$-stationary point, under standard conditions. Since the variance of the classical stochastic gradient estimator is typically large which slows down the convergence, we also apply an efficient stochastic variance-reduce proximal gradient method with an importance sampling based ProbAbilistic Gradient Estimator (PAGE). To the best of our knowledge, the application of this method represents a novel approach in addressing the general regularized reward optimization problem. Our analysis shows that the sample complexity can be improved from $O(\epsilon^{-4})$ to $O(\epsilon^{-3})$ under additional conditions. Our results on the stochastic (variance-reduced) proximal gradient method match the sample complexity of their most competitive counterparts under similar settings in the RL literature.

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with $\mathcal{C}$-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for $\mathcal{C}$-mixing sequences and the neural network approximation theory for the H\"{o}lder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.

A Finite Expression Method for Solving High-Dimensional Committor Problems

Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states $A$ and $B$. Central to TPT is the committor function, which describes the probability to hit the metastable state $B$ prior to $A$ from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.

Spectral Clustering via Orthogonalization-Free Methods

Graph Signal Filter used as dimensionality reduction in spectral clustering usually requires expensive eigenvalue estimation. We analyze the filter in an optimization setting and propose to use four orthogonalization-free methods by optimizing objective functions as dimensionality reduction in spectral clustering. The proposed methods do not utilize any orthogonalization, which is known as not well scalable in a parallel computing environment. Our methods theoretically construct adequate feature space, which is, at most, a weighted alteration to the eigenspace of a normalized Laplacian matrix. We numerically hypothesize that the proposed methods are equivalent in clustering quality to the ideal Graph Signal Filter, which exploits the exact eigenvalue needed without expensive eigenvalue estimation. Numerical results show that the proposed methods outperform Power Iteration-based methods and Graph Signal Filter in clustering quality and computation cost. Unlike Power Iteration-based methods and Graph Signal Filter which require random signal input, our methods are able to utilize available initialization in the streaming graph scenarios. Additionally, numerical results show that our methods outperform ARPACK and are faster than LOBPCG in the streaming graph scenarios. We also present numerical results showing the scalability of our methods in multithreading and multiprocessing implementations to facilitate parallel spectral clustering.

Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives

This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.

Finite Expression Methods for Discovering Physical Laws from Data

Nonlinear dynamics is a pervasive phenomenon observed in various scientific and engineering disciplines. However, uncovering analytical expressions that describe nonlinear dynamics from limited data remains a challenging and essential task. In this paper, we propose a new deep symbolic learning method called the ``finite expression method'' (FEX) to identify the governing equations within the space of functions containing a finite set of analytic expressions, based on observed dynamic data. The core idea is to leverage FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions using convolutions. Our numerical results demonstrate that FEX outperforms all existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across various problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Furthermore, the results highlight that FEX exhibits flexibility and expressive power in accurately approximating symbolic governing equations, while maintaining low memory and favorable time complexity.

Convergence Analysis of the Deep Galerkin Method for Weak Solutions

This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively. In DGMW, a deep neural network is applied to parametrize the PDE solution, and a second neural network is adopted to parametrize the test function in the traditional Galerkin formulation. By properly choosing the depth and width of these two networks in terms of the number of training samples $n$, it is shown that the convergence rate of DGMW is $\mathcal{O}(n^{-1/d})$, which is the first convergence result for weak solutions. The main idea of the proof is to divide the error of the DGMW into an approximation error and a statistical error. We derive an upper bound on the approximation error in the $H^{1}$ norm and bound the statistical error via Rademacher complexity.

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

Deep neural networks (DNNs) have seen tremendous success in many fields and their developments in PDE-related problems are rapidly growing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems. We apply our results to various PDEs, including elliptic equations, parabolic equations, and Burgers equations.