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A numerical framework based on network partition and operator splitting is developed to solve nonlinear differential equations of large-scale dynamic processes encountered in physics, chemistry and biology. Under the assumption that those…
Quantum algorithms for both differential equation solving and for machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in…
Numerical solutions of partial differential equations enable a broad range of scientific research. The Dedalus Project is a flexible, open-source, parallelized computational framework for solving general partial differential equations using…
Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently…
Differential privacy is a mathematical framework for developing statistical computations with provable guarantees of privacy and accuracy. In contrast to the privacy component of differential privacy, which has a clear mathematical and…
Underground duct banks carrying power cables dissipate heat to the surrounding soil. The amount of heat dissipated determines the current rating of cables, which in turn affects the sizing of the cables. The dissipation of heat through the…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
Neural network-based methods for solving differential equations have been gaining traction. They work by improving the differential equation residuals of a neural network on a sample of points in each iteration. However, most of them employ…
Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the…
When samples have internal structure, we often see a mismatch between the objective optimized during training and the model's goal during inference. For example, in sequence-to-sequence modeling we are interested in high-quality translated…
Earth System Models (ESMs) are the primary tools for investigating future Earth system states at time scales from decades to centuries, especially in response to anthropogenic greenhouse gas release. State-of-the-art ESMs can reproduce the…
Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep…
Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create…
The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients…
Dealing with planning problems with both logical relations and numeric changes in real-world dynamic environments is challenging. Existing numeric planning systems for the problem often discretize numeric variables or impose convex…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the…
Algorithmic differentiation (AD) allows exact computation of derivatives given only an implementation of an objective function. Although many AD tools are available, a proper and efficient implementation of AD methods is not…
High Performance Distributed Computing is essential to boost scientific progress in many areas of science and to efficiently deploy a number of complex scientific applications. These applications have different characteristics that require…
Simflowny is an open platform which automatically generates efficient parallel code of scientific dynamical models for different simulation frameworks. Here we present major upgrades on this software to support simultaneously a quite…