Related papers: Efficient Training of Physics-enhanced Neural ODEs…
Neural Ordinary Differential Equations (Neural ODEs) represent continuous-time dynamics with neural networks, offering advancements for modeling and control tasks. However, training Neural ODEs requires solving differential equations at…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs…
Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations.…
Embedding nonlinear dynamical systems into artificial neural networks is a powerful new formalism for machine learning. By parameterizing ordinary differential equations (ODEs) as neural network layers, these Neural ODEs are…
Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods…
A conventional approach to train neural ordinary differential equations (ODEs) is to fix an ODE solver and then learn the neural network's weights to optimize a target loss function. However, such an approach is tailored for a specific…
Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These…
Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from a certain initial state $\mathbf{x}_0$ to a desired target state $\mathbf{x}^*$ in finite time $T$. Recent advances…
Real-world systems are often formulated as constrained optimization problems. Techniques to incorporate constraints into Neural Networks (NN), such as Neural Ordinary Differential Equations (Neural ODEs), have been used. However, these…
In order to drastically reduce the heavy computational burden associated with time-domain simulations, this paper introduces a Physics-Informed Neural Network (PINN) to directly learn the solutions of power system dynamics. In contrast to…
Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…
In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for…
Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep…
Neural ODEs demonstrate strong performance in generative and time-series modelling. However, training them via the adjoint method is slow compared to discrete models due to the requirement of numerically solving ODEs. To speed neural ODEs…
To better understand and improve the behavior of neural networks, a recent line of works bridged the connection between ordinary differential equations (ODEs) and deep neural networks (DNNs). The connections are made in two folds: (1) View…
Training neural ODEs on large datasets has not been tractable due to the necessity of allowing the adaptive numerical ODE solver to refine its step size to very small values. In practice this leads to dynamics equivalent to many hundreds or…
Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…
Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into neural network training objectives. However, their deployment on…
A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models…