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Understanding how complex systems respond to perturbations, such as whether they will remain stable or what their most sensitive patterns are, is a fundamental challenge across science and engineering. Traditional stability and receptivity…
Recent work by Xia et al. leveraged the continuous-limit of the classical momentum accelerated gradient descent and proposed heavy-ball neural ODEs. While this model offers computational efficiency and high utility over vanilla neural ODEs,…
We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an…
The neural dynamics underlying brain activity are critical to understanding cognitive processes and mental disorders. However, current voxel-based whole-brain dimensionality reduction techniques fall short of capturing these dynamics,…
We present a framework that leverages the Discrete Empirical Interpolation Method (DEIM) for interpretable deep learning and dynamical system analysis. Although DEIM efficiently approximates nonlinear terms in projection-based reduced-order…
Mobile robots, such as ground vehicles and quadrotors, are becoming increasingly important in various fields, from logistics to agriculture, where they automate processes in environments that are difficult to access for humans. However, to…
Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…
Dynamical analysis of manufacturing and natural systems provides critical information about production of manufactured and natural resources respectively, thus playing an important role in assessing sustainability of these systems. However,…
Neural Ordinary Differential Equations (NODEs) are a new class of models that transform data continuously through infinite-depth architectures. The continuous nature of NODEs has made them particularly suitable for learning the dynamics of…
We propose a nonlinear reduced basis method for the efficient approximation of parametrized partial differential equations (PDEs), exploiting kernel proper orthogonal decomposition (KPOD) for the generation of a reduced-order space and…
We focus on learning unknown dynamics from data using ODE-nets templated on implicit numerical initial value problem solvers. First, we perform Inverse Modified error analysis of the ODE-nets using unrolled implicit schemes for ease of…
Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical…
Data-driven modeling of constrained multibody dynamics remains challenged by (i) the training cost of Neural ODEs, which typically require backpropagation through an ODE solver, and (ii) error accumulation in rollout predictions. We…
Steering a system towards a desired target in a very short amount of time is challenging from a computational standpoint. Indeed, the intrinsically iterative nature of optimal control problems requires multiple simulations of the physical…
Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For…
The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover,…
Accurately learning solution operators for time-dependent partial differential equations (PDEs) from sparse and irregular data remains a challenging task. Recurrent DeepONet extensions inherit the discrete-time limitations of…
Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…
Time-delay dynamical systems inherently embody infinite-dimensional dynamics, thereby amplifying their complexity. This aspect is especially notable in nonlinear dynamical systems, which frequently defy analytical solutions and necessitate…
The discovery of conservation laws is a cornerstone of scientific progress. However, identifying these invariants from observational data remains a significant challenge. We propose a hybrid framework to automate the discovery of conserved…