Related papers: Hierarchical Implicit Neural Emulators
In this note we propose and analyze novel implicit-explicit methods based on second order strong stability preserving multistep time discretizations. Several schemes are developed, and a linear stability analysis is performed to study their…
Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of…
We present a differentiable soft-body physics simulator that can be composed with neural networks as a differentiable layer. In contrast to other differentiable physics approaches that use explicit forward models to define state…
Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network…
One of the most popular recent areas of machine learning predicates the use of neural networks augmented by information about the underlying process in the form of Partial Differential Equations (PDEs). These physics-informed neural…
The effectiveness of recurrent neural networks can be largely influenced by their ability to store into their dynamical memory information extracted from input sequences at different frequencies and timescales. Such a feature can be…
Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with representative datasets. Recently, an augmented framework has been…
In this paper we address the question of whether it is possible to integrate time-dependent high-dimensional PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for…
Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires…
We consider the problem of forecasting complex, nonlinear space-time processes when observations provide only partial information of on the system's state. We propose a natural data-driven framework, where the system's dynamics are modelled…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
This report considers linear multistep methods through time filtering. The approach has several advantages. It is modular and requires the addition of only one line of additional code. Error estimation and variable timesteps is…
Physical systems whose dynamics are governed by partial differential equations (PDEs) find applications in numerous fields, from engineering design to weather forecasting. The process of obtaining the solution from such PDEs may be…
Numerical ice sheet models compute evolving ice geometry and velocity fields using various stress-balance approximations and boundary conditions. At high spatial resolution, with horizontal mesh/grid resolutions of a few kilometers or…
Time series forecasting based on deep architectures has been gaining popularity in recent years due to their ability to model complex non-linear temporal dynamics. The recurrent neural network is one such model capable of handling…
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…
Simulating physical problems involving multi-time scale coupling is challenging due to the need of solving these multi-time scale processes simultaneously. In response to this challenge, this paper proposed an explicit multi-time step…
Singularly perturbed systems (SPSs) are prevalent in engineering applications, where numerically solving their initial value problems (IVPs) is challenging due to stiffness arising from multiple time scales. Classical explicit methods…
In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms…
Long-horizon spatiotemporal prediction remains a challenging problem due to cumulative errors, noise amplification, and the lack of physical consistency in existing models. While diffusion models provide a probabilistic framework for…