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Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we…
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…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…
The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated…
We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control. However, the numerical analysis of FDEs has faced challenges due to its…
The Adomian decomposition method is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The aim of this paper is to apply Adomian decomposition method to obtain approximate solutions of nonlinear…
Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation is limited. In network models with delay, the delayed channels are low-dimensional and accounting for this…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. These systems cannot be properly modelled and simulated with standard…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient,…
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws…
Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to…
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of…
Neural ODEs (NODEs) have emerged as powerful tools for modeling time series data, offering the flexibility to adapt to varying input scales and capture complex dynamics. However, they face significant challenges: first, their reliance on…
Recently, it has been shown in [Jentzen, A., M\"uller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of autonomous stochastic differential equations (SDE) on the time interval $[0,T]$ with…
Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the…