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arXiv 2024年

作者： Khawaled, Samah Warfield, Simon K. Freiman, Moti The Interdisciplinary program in Applied Mathematics Faculty of Mathematics Technion-Israel Institute of Technology Faculty of Biomedical Engineering Technion-Israel Institute of Technology Computational Radiology Laboratory Boston Children's Hospital BostonMA United States Harvard Medical School BostonMA United States

Functional Magnetic Resonance Imaging (fMRI) is vital in neuroscience, enabling investigations into brain disorders, treatment monitoring, and brain function mapping. However, head motion during fMRI scans, occurring between shots of slice acquisition, can result in distortion, biased analyses, and increased costs due to the need for scan repetitions. Therefore, retrospective slice-level motion correction through slice-to-volume registration (SVR) is crucial. Previous studies have utilized deep learning (DL) based models to address the SVR task;however, they overlooked the uncertainty stemming from the input stack of slices and did not assign weighting or scoring to each slice. In this work, we introduce an end-to-end SVR model for aligning 2D fMRI slices with a 3D reference volume, incorporating a self-attention mechanism to enhance robustness against input data variations and uncertainties. It utilizes independent slice and volume encoders and a self-attention module to assign pixel-wise scores for each slice. We conducted evaluation experiments on 200 images involving synthetic rigid motion generated from 27 subjects belonging to the test set, from the publicly available Healthy Brain Network (HBN) dataset. Our experimental results demonstrate that our model achieves competitive performance in terms of alignment accuracy compared to state-of-the-art deep learning-based methods (Euclidean distance of 0.93 [mm] vs. 1.86 [mm]). Furthermore, our approach exhibits significantly faster registration speed compared to conventional iterative methods (0.096 sec. vs. 1.17 sec.). Our end-to-end SVR model facilitates real-time head motion tracking during fMRI acquisition, ensuring reliability and robustness against uncertainties in inputs. source code, which includes the training and evaluations, will be available soon. Copyright © 2024, The Authors. All rights reserved.

关键词： Iterative methods

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CVIT: CONTINUOUS VISION TRANSFORMER FOR OPERATOR LEARNING

arXiv

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arXiv 2024年

作者： Wang, Sifan Seidman, Jacob H. Sankaran, Shyam Wang, Hanwen Pappas, George J. Perdikaris, Paris Institution for Foundation of Data Science Yale University United States Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania United States Department of Electrical and Systems Engineering University of Pennsylvania United States

Operator learning, which aims to approximate maps between infinite-dimensional function spaces, is an important area in scientific machine learning with applications across various physical domains. Here we introduce the Continuous Vision Transformer (CViT), a novel neural operator architecture that leverages advances in computer vision to address challenges in learning complex physical systems. CViT combines a vision transformer encoder, a novel grid-based coordinate embedding, and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. This design allows for flexible output representations and consistent evaluation at arbitrary resolutions. We demonstrate CViT’s effectiveness across a diverse range of partial differential equation (PDE) systems, including fluid dynamics, climate modeling, and reaction-diffusion processes. Our comprehensive experiments show that CViT achieves state-of-the-art performance on multiple benchmarks, often surpassing larger foundation models, even without extensive pretraining and roll-out fine-tuning. Taken together, CViT exhibits robust handling of discontinuous solutions, multi-scale features, and intricate spatio-temporal dynamics. Our contributions can be viewed as a significant step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in the physical sciences. All data and code are publicly available at https://***/PredictiveIntelligenceLab/cvit. © 2024, CC BY-NC-SA.

关键词： Computer vision

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Machine learning from a continuous viewpoint,I

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Science China mathematics 2020年第11期63卷 2233-2266页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous *** also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous *** discuss how the issues of generalization error and implicit regularization can be studied under this framework.

关键词： machine learning continuous formulation flow-based model gradient flow particle approximation

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Perturbational Complexity by Distribution Mismatch: A Systematic Analysis of Reinforcement Learning in Reproducing Kernel Hilbert Space

arXiv

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arXiv 2021年

作者： Long, Jihao Han, Jiequn Program of Applied and Computational Mathematics Princeton University United States Center for Computational Mathematics Flatiron Institute United States

Most existing theoretical analysis of reinforcement learning (RL) is limited to the tabular setting or linear models due to the difficulty in dealing with function approximation in high dimensional space with an uncertain environment. This work offers a fresh perspective into this challenge by analyzing RL in a general reproducing kernel Hilbert space (RKHS). We consider a family of Markov decision processes M of which the reward functions lie in the unit ball of an RKHS and transition probabilities lie in a given arbitrary set. We define a quantity called perturbational complexity by distribution mismatch ∆M(ǫ) to characterize the complexity of the admissible state-action distribution space in response to a perturbation in the RKHS with scale ǫ. We show that ∆M(ǫ) gives both the lower bound of the error of all possible algorithms and the upper bound of two specific algorithms (fitted reward and fitted Q-iteration) for the RL problem. Hence, the decay of ∆M(ǫ) with respect to ǫ measures the difficulty of the RL problem on M. We further provide some concrete examples and discuss whether ∆M(ǫ) decays fast or not in these examples. As a byproduct, we show that when the reward functions lie in a high dimensional RKHS, even if the transition probability is known and the action space is finite, it is still possible for RL problems to suffer from the curse of dimensionality. Copyright © 2021, The Authors. All rights reserved.

关键词： Iterative methods

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NOMAD: nonlinear manifold decoders for operator learning 22

NOMAD: nonlinear manifold decoders for operator learning

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Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Jacob H. Seidman Georgios Kissas Paris Perdikaris George J. Pappas Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Department of Electrical and Systems Engineering University of Pennsylvania

ISBN: (纸本)9781713871088

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost.

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Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

arXiv

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arXiv 2024年

作者： Awadelkarim, Elsiddig Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia School of Data Science The Chinese University of Hong Kong Shenzhen China

In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for ϵ > 0, to obtain a mean square error of O(ϵ2) the PF has a cost per-observation time of O(ϵ−5) and the MLPF costs O(ϵ−4) (best case) or O(ϵ−4 log(ϵ)2) (worst case). Our theoretical results are supported by numerical experiments. © 2024, CC BY.

关键词： Mean square error

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Unbiased Parameter Estimation for Bayesian Inverse Problems

arXiv

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arXiv 2025年

作者： Chada, Neil K. Jasra, Ajay Maama, Mohamed Tempone, Raul Department of Mathematics City University of Hong Kong China School of Data Science The Chinese University of Hong Kong Shenzhen CN Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximizer of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

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

arXiv

引用

arXiv 2022年

作者： Han, Jiequn Hu, Ruimeng Long, Jihao Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Department of Mathematics Department of Statistics and Applied Probability University of California Santa BarbaraCA93106-3080 United States The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

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 is 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 (IPMs) 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 *** Codes 65C30, 68T07, 49N80, 68Q25 © 2022, CC BY.

关键词： Stochastic systems

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ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE

arXiv

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arXiv 2022年

作者： Greengard, Philip Hoskins, Jeremy G. Marshall, Nicholas F. Singer, Amit Department of Statistics Columbia University United States Department of Statistics University of Chicago United States Program in Applied and Computational Mathematics The Department of Mathematics Princeton University United States

This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W2. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = -∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1]2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented. Copyright © 2022, The Authors. All rights reserved.

关键词： Laplace transforms

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STEFAN PROBLEM WITH SURFACE TENSION: GLOBAL EXISTENCE OF PHYSICAL SOLUTIONS UNDER RADIAL SYMMETRY

arXiv

引用

arXiv 2022年

作者： Nadtochiy, Sergey Shkolnikov, Mykhaylo Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University Princeton NJ08544 United States

We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d ⩾ 3, in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface *** Codes 80A22, 60H30, 35B44 Copyright © 2022, The Authors. All rights reserved.

关键词： Surface tension

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