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Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer-generated imagery to computer-aided design in research and development. However, solving the partial differential equations of…
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…
We present a stochastic method for reconstructing missing spatial and velocity data along the trajectories of small objects passively advected by turbulent flows with a wide range of temporal or spatial scales, such as small balloons in the…
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…
The paper presents a versatile framework for solids which undergo nonisothermal processes with irreversibly changing microstructure at large strains. It outlines rate-type and incremental variational principles for the full thermomechanical…
We introduce the concept of Randomly Modulated Gaussian Processes as a unifying framework for modeling, analyzing and classifying anomalous diffusion models in heterogeneous media. This formulation incorporates correlations in the…
Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes…
Traditional elastic wavefield separation methods, while accurate, often demand substantial computational resources, especially for large geological models or 3D scenarios. Purely data-driven neural network approaches can be more efficient,…
Existing deterministic variational inference approaches for diffusion processes use simple proposals and target the marginal density of the posterior. We construct the variational process as a controlled version of the prior process and…
We investigate non-equilibrium critical phenomena using a nonperturbative renormalization group method. Reaction-diffusion processes are described by a scale dependent effective action which evolution is governed by very generic flow…
This paper explores the challenges and benefits of a trainable destruction process in diffusion samplers -- diffusion-based generative models trained to sample an unnormalised density without access to data samples. Contrary to the majority…
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have…
Conventional approaches for simulating steady-state distributions of particles under diffusive and advective transport at high P\'eclet numbers involve solving the diffusion and advection equations in at least two dimensions. Here, we…
Time series generation focuses on modeling the underlying data distribution and resampling to produce authentic time series data. Key components, such as trend and seasonality, drive temporal fluctuations, yet many existing approaches fail…
We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the…
These are lecture notes for various Summer and Winter schools that I have given. The notes describe the methodology called Variational Modelling, and focus on the application to the modelling of gradient-flow systems. I describe the…
Learning nonlinear dynamics from diffusion data is a challenging problem since the individuals observed may be different at different time points, generally following an aggregate behaviour. Existing work cannot handle the tasks well since…
We propose a conditional stochastic interpolation (CSI) method for learning conditional distributions. CSI is based on estimating probability flow equations or stochastic differential equations that transport a reference distribution to the…
We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this second…
The complex dynamics of physical systems can often be modeled with stochastic differential equations. However, computational constraints inhibit the estimation of dynamics from large time-series datasets. I present a method for estimating…