Related papers: A machine learning framework for data driven accel…
We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient…
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws…
This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order…
Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise…
Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…
We propose a data-driven framework for learning reduced-order moment dynamics from PDE-governed systems using Neural ODEs. In contrast to derivative-based methods like SINDy, which necessitate densely sampled data and are sensitive to…
Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…
There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…
Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on…
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…
We consider chance-constrained problems with discrete random distribution. We aim for problems with a large number of scenarios. We propose a novel method based on the stochastic gradient descent method which performs updates of the…
We introduce a machine-learning framework to learn the hyperparameter sequence of first-order methods (e.g., the step sizes in gradient descent) to quickly solve parametric convex optimization problems. Our computational architecture…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
A surrogate model approximates the outputs of a solver of Partial Differential Equations (PDEs) with a low computational cost. In this article, we propose a method to build learning-based surrogates in the context of parameterized PDEs,…
A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based…
Pairwise learning, an important domain within machine learning, addresses loss functions defined on pairs of training examples, including those in metric learning and AUC maximization. Acknowledging the quadratic growth in computation…
We present a new approach for estimating parameters in rational ODE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a…
We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive…
Advances in differentiable numerical integrators have enabled the use of gradient descent techniques to learn ordinary differential equations (ODEs). In the context of machine learning, differentiable solvers are central for Neural ODEs…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…