相关论文: Partial and complete linearization of PDEs based o…
In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential…
Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents…
For nonlinear Schr\"odinger equations in less than or equal to four dimension, with non-vanishing initial data at infinity, a new approach to derive the conservation law is obtained. Since this approach does not contain approximating…
This paper develops methods for simplifying systems of partial differential equations that have families of conservation laws which depend on functions of the independent or dependent variables. In some cases, such methods can be combined…
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation…
Approximate nonlinear self-adjointness is an effective method to construct approximate conservation law of perturbed partial differential equations (PDEs). In this paper, we study the relations between approximate nonlinear self-adjointness…
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…
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable…
Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of…
We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a non-linear generalization…
Carleman linearization is a mathematical technique that transforms nonlinear dynamical systems into infinite-dimensional linear systems, enabling simplified analysis. Initially developed for ordinary differential equations (ODEs) and later…
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…
Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify,…
The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…
A complete classification of all low-order conservation laws is carried out for a system of coupled semilinear wave equations which is a natural two-component generalization of the nonlinear Klein-Gordon equation. The conserved quantities…
We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns:…
We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial data. Based on an exact mapping between nonlinear and linear PDEs using the level set method, these new quantum algorithms for nonlinear…
Multi-query applications such as parameter estimation, uncertainty quantification and design optimization for parameterized PDE systems are expensive due to the high computational cost of high-fidelity simulations. Reduced/Latent state…
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…