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Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy…

Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…

In recent years Deep Neural Networks (DNNs) have been rapidly developed in various applications, together with increasingly complex architectures. The performance gain of these DNNs generally comes with high computational costs and large…

Uncertainty quantification (UQ) helps to make trustworthy predictions based on collected observations and uncertain domain knowledge. With increased usage of deep learning in various applications, the need for efficient UQ methods that can…

Neural Stochastic Differential Equations (NSDEs) model the drift and diffusion functions of a stochastic process as neural networks. While NSDEs are known to make accurate predictions, their uncertainty quantification properties have been…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Neural networks (NNs) can achieved high performance in various fields such as computer vision, and natural language processing. However, deploying NNs in resource-constrained safety-critical systems has challenges due to uncertainty in the…

Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it…

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…

The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this…

Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex…

We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The…

The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution…

In the past years, the application of neural networks as an alternative to classical numerical methods to solve Partial Differential Equations has emerged as a potential paradigm shift in this century-old mathematical field. However, in…

Calibration of neural networks is a topical problem that is becoming more and more important as neural networks increasingly underpin real-world applications. The problem is especially noticeable when using modern neural networks, for which…

In this paper, we propose a deep learning-based method, deep Euler method (DEM) to solve ordinary differential equations. DEM significantly improves the accuracy of the Euler method by approximating the local truncation error with deep…

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…