Related papers: Solving Differential Equations Using Neural Networ…

Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present…

We propose an approach to solving partial differential equations (PDEs) using a set of neural networks which we call Neural Basis Functions (NBF). This NBF framework is a novel variation of the POD DeepONet operator learning approach where…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the…

Recently, a lot of papers proposed to use neural networks to approximately solve partial differential equations (PDEs). Yet, there has been a lack of flexible framework for convenient experimentation. In an attempt to fill the gap, we…

Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the…

Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…

Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for…

Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or…

Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a…

Continuous deep learning models, referred to as Neural Ordinary Differential Equations (Neural ODEs), have received considerable attention over the last several years. Despite their burgeoning impact, there is a lack of formal analysis…

Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of…

The fast simulation of dynamical systems is a key challenge in many scientific and engineering applications, such as weather forecasting, disease control, and drug discovery. With the recent success of deep learning, there is increasing…

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…

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…

Solving partial differential equations (PDEs) is the cornerstone of scientific research and development. Data-driven machine learning (ML) approaches are emerging to accelerate time-consuming and computation-intensive numerical simulations…

Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here…

Neural ordinary differential equations (NODE) have been proposed as a continuous depth generalization to popular deep learning models such as Residual networks (ResNets). They provide parameter efficiency and automate the model selection…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

We introduce a provably stable variant of neural ordinary differential equations (neural ODEs) whose trajectories evolve on an energy functional parametrised by a neural network. Stable neural flows provide an implicit guarantee on…