Related papers: Hybrid solver methods for ODEs: Machine-Learning c…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
Recent machine learning algorithms dedicated to solving semi-linear PDEs are improved by using different neural network architectures and different parameterizations. These algorithms are compared to a new one that solves a fixed point…
Ordinary Differential Equations (ODEs) are widely used in physics, chemistry, and biology to model dynamic systems, including reaction kinetics, population dynamics, and biological processes. In this work, we integrate GPU-accelerated ODE…
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…
Regularization-based approaches for injecting constraints in Machine Learning (ML) were introduced to improve a predictive model via expert knowledge. We tackle the issue of finding the right balance between the loss (the accuracy of the…
Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…
This paper surveys the trend of leveraging machine learning to solve mixed integer programming (MIP) problems. Theoretically, MIP is an NP-hard problem, and most of the combinatorial optimization (CO) problems can be formulated as the MIP.…
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 technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
In this article, we discuss the numerical solution of Boolean polynomial programs by algorithms borrowing from numerical methods for differential equations, namely the Houbolt scheme, the Lie scheme, and a Runge-Kutta scheme. We first…
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…
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…
Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) that is identical to that of the Gradient Descent Ascent (GDA) method when derived naively. However,…
The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved…
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the…
This work focuses on the numerical study of a recently published class of Runge-Kutta methods designed for mixed-precision arithmetic. We employ the methods in solving partial differential equations on modern hardware. In particular we…