Related papers: Bridging Traditional and Machine Learning-based Al…
Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for…
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…
Kernel methods are powerful and flexible approach to solve many problems in machine learning. Due to the pairwise evaluations in kernel methods, the complexity of kernel computation grows as the data size increases; thus the applicability…
We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
Partial differential equations (PDEs) with multiple scales or those defined over sufficiently large domains arise in various areas of science and engineering and often present problems when approximating the solutions numerically. Machine…
We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping…
Kernel learning methods are among the most effective learning methods and have been vigorously studied in the past decades. However, when tackling with complicated tasks, classical kernel methods are not flexible or "rich" enough to…
The simulation of complex systems, such as gas transport in large pipeline networks, often involves solving PDEs posed on intricate graph structures. Such problems require considerable computational and memory resources. The Random Batch…
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…
Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance,…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…
We present a novel method for using Neural Networks (NNs) for finding solutions to a class of Partial Differential Equations (PDEs). Our method builds on recent advances in Neural Radiance Field research (NeRFs) and allows for a NN to…
In multi-view medical diagnosis, deep learning-based models often fuse information from different imaging perspectives to improve diagnostic performance. However, existing approaches are prone to overfitting and rely heavily on…
Among the various machine learning methods solving partial differential equations, the Random Feature Method (RFM) stands out due to its accuracy and efficiency. In this paper, we demonstrate that the approximation error of RFM exhibits…
Robust parameter estimation is a crucial task in several 3D computer vision pipelines such as Structure from Motion (SfM). State-of-the-art algorithms for robust estimation, however, still suffer from difficulties in converging to…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…