Related papers: Training Stiff Neural Ordinary Differential Equati…
The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential algebraic equation system into a finitely stiff ordinary differential equation (ODE)…
Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
We propose a novel approach for training Physics-enhanced Neural ODEs (PeN-ODEs) by expressing the training process as a dynamic optimization problem. The full model, including neural components, is discretized using a high-order implicit…
The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often…
Euler--Euler or volume-averaged Navier--Stokes equations are used in various applications to model systems with two or more interpenetrating phases. Each fluid obeys its own momentum and mass equations, and the phases are typically coupled…
A quantitative definition of numerical stiffness for initial value problems is proposed. Exponential integrators can effectively integrate linearly stiff systems, but they become expensive when the linear coefficient is a matrix, especially…
A method for enhancing the stability and robustness of explicit schemes in computational fluid dynamics is presented. The method is based in reformulating explicit schemes in matrix form, which cane modified gradually into semi or…
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…
Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states,…
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of…
We introduce a class of explicit exponential Rosenbrock methods for the time integration of large systems of stiff differential equations. Their application with respect to simulation tasks in the field of visual computing is discussed…
The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and…
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A…
We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion is also, to a certain extent,…
We propose a simple interpolation-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method (known in the literature as "adjoint method") to train neural ODEs on…
In this paper, we present error estimates of the integral deferred correction method constructed with stiffly accurate implicit Runge-Kutta methods with a nonsingular matrix $A$ in its Butcher table representation, when applied to stiff…
We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical solution non-expansive whenever the exact solution is…
The main difficulty in studying numerical method for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (J. Printems. [ESAIM Math. Model. Numer. Anal. (2001)]). Although fruitful results on numerical…
In this paper we present a formally fourth-order accurate hybrid-variable method for the Euler equations in the context of method of lines. The hybrid-variable (HV) method seeks numerical approximations to both cell-averages and nodal…