相关论文: Computation of Partially Invariant Solutions for t…

We propose an effective and robust algorithm for identifying partial differential equations (PDEs) with space-time varying coefficients from a single trajectory of noisy observations. Identifying unknown differential equations from noisy…

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point…

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…

Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems…

Dimensionality reduction algorithms like principal component analysis (PCA) are workhorses of machine learning and neuroscience, but each has well-known limitations. Variants of PCA are simple and interpretable, but not flexible enough to…

The explicit governing equation is one of the simplest and most intuitive forms for characterizing physical laws. However, directly discovering partial differential equations (PDEs) from data poses significant challenges, primarily in…

In this paper, we propose relative projective differential invariants (RPDIs) which are invariant to general projective transformations. By using RPDIs and the structural frame of integral invariant, projective weighted moment invariants…

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…

Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their…

Invariant Coordinate Selection (ICS) is a multivariate data transformation and a dimension reduction method that can be useful in many different contexts. It can be used for outlier detection or cluster identification, and can be seen as an…

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…

In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…

We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…

Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that lead to fulfilling a PDE can be challenging and…

We discuss a hierarchy of broken symmetries with special emphasis on partial dynamical symmetries (PDS). The latter correspond to a situation in which a non-invariant Hamiltonian accommodates a subset of solvable eigenstates with good…

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for…

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…

We show that matrix $Q\times Q$ Self-dual type $S$-integrable Partial Differential Equations (PDEs) possess a family of lower-dimensional reductions represented by the matrix $ Q \times n_0 Q$ quasilinear first order PDEs solved in…

We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. The new method can easily be combined with a finite…

In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We apply this approach to…