Related papers: Computer-assisted proofs in PDE: a survey
Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy…
We compare the values associated with (traditional) community based proof verification to those associated with computer proof verification. We propose ways that computer proofs might incorporate successful strategies from human…
We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…
Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to…
The advent of electronic computers has revolutionised the application of statistical mechanics to the liquid state. Computers have permitted, for example, the calculation of the phase diagram of water and ice and the folding of proteins.…
THis work is a survey of a few nonlinear PDE based models in image restoring.
First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general…
Analog computers can be revived as a feasible technology platform for low precision, energy efficient and fast computing. We justify this statement by measuring the performance of a modern analog computer and comparing it with that of…
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this…
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…
A physics-informed neural network (PINN), which has been recently proposed by Raissi et al [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
A computational method for numeric resolution of a PDEs system, based on a Finite Differences schema integrated by interpolations of partial results, and an estimate of the error of its solution respect to the normal FD solution.
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…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
Proof competence, i.e. the ability to write and check (mathematical) proofs, is an important skill in Computer Science, but for many students it represents a difficult challenge. The main issues are the correct use of formal language and…
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 application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…