The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks

Sung Woong Cho; Sunwoo Hwang; Hyung Ju Hwang; Sung Woong Cho; Sunwoo Hwang; Hyung Ju Hwang

doi:10.3934/mbe.2023309

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 7154-7170. doi: 10.3934/mbe.2023309

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The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks

Department of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea

Academic Editor: Zhi-An Wang

Received: 31 October 2022 Revised: 19 January 2023 Accepted: 01 February 2023 Published: 10 February 2023

In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.
- partial differential equation,
- traveling wave solution,
- forward and inverse problems,
- physics-informed neural networks,
- monotonic neural networks,
- convergence analysis
Citation: Sung Woong Cho, Sunwoo Hwang, Hyung Ju Hwang. The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7154-7170. doi: 10.3934/mbe.2023309

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Abstract

In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.

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