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Data-driven modeling of dynamical systems often faces numerous data-related challenges. A fundamental requirement is the existence of a unique set of parameters for a chosen model structure, an issue commonly referred to as identifiability.…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…
One of the outstanding analytical problems in X-ray single particle imaging (SPI) is the classification of structural heterogeneity, which is especially difficult given the low signal-to-noise ratios of individual patterns and that even…
We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the…
Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the discovery problem. Recently, the discovery of traditional ODEs dynamics using linear multistep methods…
Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of these systems' stochastic…
Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such as estimating…
We present a framework designed to learn the underlying dynamics between two images observed at consecutive time steps. The complex nature of image data and the lack of temporal information pose significant challenges in capturing the…
Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified…
The ability to discover physical laws and governing equations from data is one of humankind's greatest intellectual achievements. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations (PDEs), these emergent patterns sometimes appear as local minimisers of a…
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In…
There has been a surge in the interest of using machine learning techniques to assist in the scientific process of formulating knowledge to explain observational data. We demonstrate the use of Bayesian Hidden Physics Models to first…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
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…
A systematic Bayesian framework is developed for physics constrained parameter inference ofstochastic differential equations (SDE) from partial observations. The physical constraints arederived for stochastic climate models but are…
This work applies concepts of artificial neural networks to identify the parameters of a mathematical model based on phase fields for damage and fracture. Damage mechanics is the part of the continuum mechanics that models the effects of…