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Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key…

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…

Many real-world systems are modelled using complex ordinary differential equations (ODEs). However, the dimensionality of these systems can make them challenging to analyze. Dimensionality reduction techniques like Proper Orthogonal…

Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In…

The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often…

We propose a novel approach for training Physics-enhanced Neural ODEs (PeN-ODEs) by expressing the training process as a dynamic optimization problem. The full model, including neural components, is discretized using a high-order implicit…

Recently, using neural networks to simulate spatio-temporal dynamics has received a lot of attention. However, most existing methods adopt pure data-driven black-box models, which have limited accuracy and interpretability. By combining…

Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…

Training deep neural networks may be challenging in real world data. Using models as black-boxes, even with transfer learning, can result in poor generalization or inconclusive results when it comes to small datasets or specific…

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…

Model-based design of experiments (MBDOE) is essential for efficient parameter estimation in nonlinear dynamical systems. However, conventional adaptive MBDOE requires costly posterior inference and design optimization between each…

Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from…

Neural ordinary differential equations (neural ODEs) can effectively learn dynamical systems from time series data, but their behavior on graph-structured data remains poorly understood, especially when applied to graphs with different size…

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

High-quality ordinary differential equation (ODE) solver libraries have a long history, going back to the 1970s. Over the past several years we have implemented, on top of the PETSc linear and nonlinear solver package, a new…

Measurement noise is an integral part while collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We…

In this paper, we implement Neural Ordinary Differential Equations in a Variational Autoencoder setting for generative time series modeling. An object-oriented approach to the code was taken to allow for easier development and research and…

Learning is a fundamental property of intelligent systems, observed across biological organisms and engineered systems. While modern intelligent systems typically rely on gradient descent for learning, the need for exact gradients and…

Online reinforcement learning (RL) is increasingly used for realizing adaptive systems in the presence of design time uncertainty. Online RL facilitates learning from actual operational data and thereby leverages feedback only available at…