Related papers: Stable Implementation of Probabilistic ODE Solvers
Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here…
We consider quadrature formulas of high order in time based on Radau-type, L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
Using existing, forward-in-time integration schemes, we demonstrate that it is possible to compute unstable, saddle-type fixed points of stiff systems of ODEs when the stable compenents are fast (i.e., rapidly damped) while the unstable…
Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end,…
We present a new Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in which it is possible to use convex optimization to perform stability analysis with little or no conservatism. The first result gives…
We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be…
ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
A subroutine for very-high-precision numerical solution of a class of ordinary differential equations is provided. For given evaluation point and equation parameters the memory requirement scales linearly with precision $P$, and the number…
We study numerical (in)stability of the Method of characteristics (MoC) applied to a system of non-dissipative hyperbolic partial differential equations (PDEs) with periodic boundary conditions. We consider three different solvers along the…
Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the…
The paper focuses on the numerical stability and accuracy of implicit time-domain integration (TDI) methods when applied for the solution of a power system model impacted by time delays. Such a model is generally formulated as a set of…
The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state weakly interacting N-particle systems. Local Lyapunov…
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution…
Ordinary Differential Equations are widespread tools to model chemical, physical, biological process but they usually rely on parameters which are of critical importance in terms of dynamic and need to be estimated directly from the data.…
Errors due to hardware or low level software problems, if detected, can be fixed by various schemes, such as recomputation from a checkpoint. Silent errors are errors in application state that have escaped low-level error detection. At…
Partial Integral Equations (PIEs) have been used to represent both systems with delay and systems of Partial Differential Equations (PDEs) in one or two spatial dimensions. In this paper, we show that these results can be combined to obtain…