Mean field games (MFG) and mean control (MFC) are critical classes of multiagent models for the efficient analysis massive populations interacting agents. Their areas application span topics in economics, finance, game theory, industrial engineering, crowd motion, more. In this paper, we provide a flexible machine learning framework numerical solution potential MFG MFC models. State-of-the-art methods solving such problems utilize spatial discretization that leads to curse dimensionality. We...

Significance Mean-field games (MFGs) is an emerging field that models large populations of agents. They play a central role in many disciplines, such as economics, data science, and engineering. Since applications come the form high-dimensional stochastic MFGs, numerical methods use spatial grids are prone to curse dimensionality. To this end, we exploit variational structure potential MFGs reformulate it generative adversarial network (GAN) training problem. This reformulation allays bit...

A normalizing flow is an invertible mapping between arbitrary probability distribution and a standard normal distribution; it can be used for density estimation statistical inference. Computing the follows change of variables formula thus requires invertibility efficient way to compute determinant its Jacobian. To satisfy these requirements, flows typically consist carefully chosen components. Continuous (CNFs) are mappings obtained by solving neural ordinary differential equation (ODE). The...

A promising trend in deep learning replaces traditional feedforward networks with implicit networks. Unlike networks, solve a fixed point equation to compute inferences. Solving for the varies complexity, depending on provided data and an error tolerance. Importantly, may be trained memory costs stark contrast whose requirements scale linearly depth. However, there is no free lunch --- backpropagation through often requires solving costly Jacobian-based arising from function theorem. We...

We propose a neural network (NN) approach that yields approximate solutions for high-dimensional optimal control (OC) problems and demonstrate its effectiveness using examples from multiagent path finding. Our in feedback form, where the policy function is given by an NN. In particular, we fuse Hamilton–Jacobi–Bellman (HJB) Pontryagin maximum principle (PMP) approaches parameterizing value with enables us to obtain approximately OCs real time without having solve optimization problem. Once...

Abstract In this work, we consider a novel inverse problem in mean-field games (MFGs). We aim to recover the MFG model parameters that govern underlying interactions among population based on limited set of noisy partial observations dynamics under aperture. Due its severe ill-posedness, obtaining good quality reconstruction is very difficult. Nonetheless, it vital stably and efficiently order uncover causes for practical needs. Our work focuses simultaneous recovery running cost interaction...

A normalizing flow (NF) is a mapping that transforms chosen probability distribution to normal distribution. Such flows are common technique used for data generation and density estimation in machine learning science. The estimate obtained with NF requires change of variables formula involves the computation Jacobian determinant transformation. In order tractably compute this determinant, continuous (CNF) its using neural ODE. Optimal transport (OT) theory has been successfully assist...

First-order optimization algorithms are widely used today. Two standard building blocks in these proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit formulas known only limited classes functions. We provide an algorithm, HJ-Prox, accurately approximating such proximals. This is derived from collection relations between proximals, Moreau envelopes, Hamilton–Jacobi (HJ) equations, heat Monte Carlo sampling. In particular,...

Abstract Inverse problems consist of recovering a signal from collection noisy measurements. These can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect data perturbations. Hand-chosen analytic yield desirable theoretical guarantees, but such approaches have limited effectiveness signals due their inability leverage large amounts available data. To this end, work fuses data-driven convex in...

Ptychography is a popular imaging technique that combines diffractive with scanning microscopy. The consists of coherent beam scanned across an object in series overlapping positions, leading to reliable and improved reconstructions. Ptychographic microscopes allow for large fields be imaged at high resolution the cost additional computational expense. In this work, we propose multigrid-based optimization framework reduce burdens large-scale ptychographic phase retrieval. Our proposed method...

We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent problems with obstacle and collision avoidance. These immediately become high-dimensional, even moderate phase-space dimensions per agent. Our fuses the Pontryagin Maximum Principle Hamilton-Jacobi-Bellman (HJB) approaches parameterizes value function network. yields controls in feedback form quick calculation robustness to disturbances system. train our model...

Inverse problems consist of recovering a signal from collection noisy measurements. These are typically cast as optimization problems, with classic approaches using data fidelity term and an analytic regularizer that stabilizes recovery. Recent plug-and-play (PnP) works propose replacing the operator for regularization in methods by data-driven denoiser. schemes obtain state-of-the-art results, but at cost limited theoretical guarantees. To bridge this gap, we present new algorithm takes...

We consider a global variable consensus alternating direction method of multipliers (ADMM) algorithm for estimating parameters partial differential equations (PDEs) asynchronously and in parallel. Motivated by problems with many measurements, we partition the data distribute resulting subproblems among available workers. Since each subproblem can be associated different forward models right-hand sides, this provides ample options tailoring to applications, including multisource multiphysics...

Indecipherable black boxes are common in machine learning (ML), but applications increasingly require explainable artificial intelligence (XAI). The core of XAI is to establish transparent and interpretable data-driven algorithms. This work provides concrete tools for situations where prior knowledge must be encoded untrustworthy inferences flagged. We use the "learn optimize" (L2O) methodology wherein each inference solves a optimization problem. Our L2O models straightforward implement,...

We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solved efficiently. In particular, each iteration ADMM-Softmax consists a linear least-squares problem, set independent small-scale smooth, convex problems, trivial dual variable update....

We present PNKH-B, a projected Newton-Krylov method for iteratively solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, the (approximate) Hessian only available through matrix-vector products. This commonly case parameter estimation, machine learning, image processing. In each iteration, uses low-rank approximation of to determine search direction construct metric used line search. The...