A Neuro-Symbolic Method for Solving Differential and Functional
Equations
- URL: http://arxiv.org/abs/2011.02415v1
- Date: Wed, 4 Nov 2020 17:13:25 GMT
- Title: A Neuro-Symbolic Method for Solving Differential and Functional
Equations
- Authors: Maysum Panju, Ali Ghodsi
- Abstract summary: We introduce a method for generating symbolic expressions to solve differential equations.
Unlike existing methods, our system does not require learning a language model over symbolic mathematics.
We show how the system can be effortlessly generalized to find symbolic solutions to other mathematical tasks.
- Score: 6.899578710832262
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: When neural networks are used to solve differential equations, they usually
produce solutions in the form of black-box functions that are not directly
mathematically interpretable. We introduce a method for generating symbolic
expressions to solve differential equations while leveraging deep learning
training methods. Unlike existing methods, our system does not require learning
a language model over symbolic mathematics, making it scalable, compact, and
easily adaptable for a variety of tasks and configurations. As part of the
method, we propose a novel neural architecture for learning mathematical
expressions to optimize a customizable objective. The system is designed to
always return a valid symbolic formula, generating a useful approximation when
an exact analytic solution to a differential equation is not or cannot be
found. We demonstrate through examples how our method can be applied on a
number of differential equations, often obtaining symbolic approximations that
are useful or insightful. Furthermore, we show how the system can be
effortlessly generalized to find symbolic solutions to other mathematical
tasks, including integration and functional equations.
Related papers
- Differentiable Programming for Differential Equations: A Review [36.67198631261628]
Differentiable programming is a cornerstone of modern scientific computing.
Differentiating functions based on the numerical solution of differential equations is non-trivial.
We provide a comprehensive review of existing techniques to compute derivatives of numerical solutions of differential equations.
arXiv Detail & Related papers (2024-06-14T03:54:25Z) - deepFDEnet: A Novel Neural Network Architecture for Solving Fractional
Differential Equations [0.0]
In each fractional differential equation, a deep neural network is used to approximate the unknown function.
The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.
arXiv Detail & Related papers (2023-09-14T12:58:40Z) - Towards true discovery of the differential equations [57.089645396998506]
Differential equation discovery is a machine learning subfield used to develop interpretable models.
This paper explores the prerequisites and tools for independent equation discovery without expert input.
arXiv Detail & Related papers (2023-08-09T12:03:12Z) - Stochastic Scaling in Loss Functions for Physics-Informed Neural
Networks [0.0]
Trained neural networks act as universal function approximators, able to numerically solve differential equations in a novel way.
Variations on traditional loss function and training parameters show promise in making neural network-aided solutions more efficient.
arXiv Detail & Related papers (2022-08-07T17:12:39Z) - D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations.
We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
arXiv Detail & Related papers (2022-06-21T17:59:20Z) - Automated differential equation solver based on the parametric
approximation optimization [77.34726150561087]
The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation.
It allows solving the wide class of equations in an automated manner without the algorithm's parameters change.
arXiv Detail & Related papers (2022-05-11T10:06:47Z) - Message Passing Neural PDE Solvers [60.77761603258397]
We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators.
We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes.
We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
arXiv Detail & Related papers (2022-02-07T17:47:46Z) - Meta-Solver for Neural Ordinary Differential Equations [77.8918415523446]
We investigate how the variability in solvers' space can improve neural ODEs performance.
We show that the right choice of solver parameterization can significantly affect neural ODEs models in terms of robustness to adversarial attacks.
arXiv Detail & Related papers (2021-03-15T17:26:34Z) - Computational characteristics of feedforward neural networks for solving
a stiff differential equation [0.0]
We study the solution of a simple but fundamental stiff ordinary differential equation modelling a damped system.
We show that it is possible to identify preferable choices to be made for parameters and methods.
Overall we extend the current literature in the field by showing what can be done in order to obtain reliable and accurate results by the neural network approach.
arXiv Detail & Related papers (2020-12-03T12:22:24Z) - Symbolically Solving Partial Differential Equations using Deep Learning [5.1964883240501605]
We describe a neural-based method for generating exact or approximate solutions to differential equations.
Unlike other neural methods, our system returns symbolic expressions that can be interpreted directly.
arXiv Detail & Related papers (2020-11-12T22:16:03Z) - The data-driven physical-based equations discovery using evolutionary
approach [77.34726150561087]
We describe the algorithm for the mathematical equations discovery from the given observations data.
The algorithm combines genetic programming with the sparse regression.
It could be used for governing analytical equation discovery as well as for partial differential equations (PDE) discovery.
arXiv Detail & Related papers (2020-04-03T17:21:57Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.