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1. Residual-based a posteriori error estimates of mixed methods for a three-field Biot’s consolidation model

   https://doi.org/10.1093/imanum/draa074  

   Li, Yuwen; Zikatanov, Ludmil T (October 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We present residual-based a posteriori error estimates of mixed finite element methods for the three-field formulation of Biot’s consolidation model. The error estimator are upper and lower bounds of the space-time discretization error up to data oscillation. As a by-product, we also obtain a new a posteriori error estimate of mixed finite element methods for the heat equation. 

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2. A machine learning framework for solving high-dimensional mean field game and mean field control problems

   https://doi.org/10.1073/pnas.1922204117  

   Ruthotto, Lars; Osher, Stanley J.; Li, Wuchen; Nurbekyan, Levon; Fung, Samy Wu (April 2020, Proceedings of the National Academy of Sciences)

   Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton–Jacobi–Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods. 

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3. Online Learning to Transport via the Minimal Selection Principle

   Guo, Wenxuan; Hur, YoonHaeng; Liang, Tengyuan; Ryan, Christopher T (July 2022, Proceedings of Machine Learning Research)

   Motivated by robust dynamic resource allocation in operations research, we study the Online Learning to Transport (OLT) problem where the decision variable is a probability measure, an infinite-dimensional object. We draw connections between online learning, optimal transport, and partial differential equations through an insight called the minimal selection principle, originally studied in the Wasserstein gradient flow setting by Ambrosio et al. (2005). This allows us to extend the standard online learning framework to the infinite-dimensional setting seamlessly. Based on our framework, we derive a novel method called the minimal selection or exploration (MSoE) algorithm to solve OLT problems using mean-field approximation and discretization techniques. In the displacement convex setting, the main theoretical message underpinning our approach is that minimizing transport cost over time (via the minimal selection principle) ensures optimal cumulative regret upper bounds. On the algorithmic side, our MSoE algorithm applies beyond the displacement convex setting, making the mathematical theory of optimal transport practically relevant to non-convex settings common in dynamic resource allocation. 

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4. Self-Consistent Velocity Matching of Probability Flows

   Li, Lingxiao; Hurault, Samuel; Solomon, Justin (December 2023, NeurIPS)

   We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives. 

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5. Adaptive Discretization in Online Reinforcement Learning

   https://doi.org/10.1287/opre.2022.2396  

   Sinclair, Sean R.; Banerjee, Siddhartha; Yu, Christina Lee (November 2022, Operations Research)

   Discretization-based approaches to solving online reinforcement learning problems are studied extensively on applications such as resource allocation and cache management. The two major questions in designing discretization-based algorithms are how to create the discretization and when to refine it. There are several experimental results investigating heuristic approaches to these questions but little theoretical treatment. In this paper, we provide a unified theoretical analysis of model-free and model-based, tree-based adaptive hierarchical partitioning methods for online reinforcement learning. We show how our algorithms take advantage of inherent problem structure by providing guarantees that scale with respect to the “zooming” instead of the ambient dimension, an instance-dependent quantity measuring the benignness of the optimal [Formula: see text] function. Many applications in computing systems and operations research require algorithms that compete on three facets: low sample complexity, mild storage requirements, and low computational burden for policy evaluation and training. Our algorithms are easily adapted to operating constraints, and our theory provides explicit bounds across each of the three facets. Funding: This work is supported by funding from the National Science Foundation [Grants ECCS-1847393, DMS-1839346, CCF-1948256, and CNS-1955997] and the Army Research Laboratory [Grant W911NF-17-1-0094]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2396 . 

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