Related papers: Automating the Design of Multigrid Methods with Ev…
The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid…
Many scientific problems focus on observed patterns of change or on how to design a system to achieve particular dynamics. Those problems often require fitting differential equation models to target trajectories. Fitting such models can be…
Designing optimal prompts and reasoning processes for large language models (LLMs) on domain-specific tasks is both necessary and challenging in real-world applications. Determining how to integrate domain knowledge, enhance reasoning…
An important component of any generation system is the mapping dictionary, a lexicon of elementary semantic expressions and corresponding natural language realizations. Typically, labor-intensive knowledge-based methods are used to…
The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…
Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in…
Combining large language models with evolutionary computation algorithms represents a promising research direction leveraging the remarkable generative and in-context learning capabilities of LLMs with the strengths of evolutionary…
Finding the optimal parameter setting (i.e. the optimal population size, the optimal mutation probability, the optimal evolutionary model etc) for an Evolutionary Algorithm (EA) is a difficult task. Instead of evolving only the parameters…
Most of the real-world problems are multimodal in nature that consists of multiple optimum values. Multimodal optimization is defined as the process of finding multiple global and local optima (as opposed to a single solution) of a…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the…
We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…
Deep generative models have been wildly successful at learning coherent latent representations for continuous data such as video and audio. However, generative modeling of discrete data such as arithmetic expressions and molecular…
This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…
This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary…
We propose a novel method for automatic program synthesis. P-Tree Programming represents the program search space through a single probabilistic prototype tree. From this prototype tree we form program instances which we evaluate on a given…
A deductive program synthesis tool takes a specification as input and derives a program that satisfies the specification. The drawback of this approach is that search spaces for such correct programs tend to be enormous, making it difficult…
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…
Fully realizing the potential of multigrid solvers often requires custom algorithms for a given application model, discretizations and even regimes of interest, despite considerable effort from the applied math community to develop fully…
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…