Data-driven forward-inverse problems for the variable coefficients Hirota equation using deep learning method

TL;DR

This paper presents an improved physics-informed neural network approach to solve forward-inverse problems for the variable coefficients Hirota equation, successfully recovering soliton solutions and unknown parameters under noisy conditions.

Contribution

The paper introduces an enhanced IPINN algorithm with adaptive activation, slope recovery, and regularization, enabling accurate data-driven solutions and parameter discovery for the VCH equation.

Findings

Successful learning of soliton solutions with network adjustments

Stable and accurate parameter training with regularization

Verification of IPINN effectiveness for forward-inverse problems

Abstract

Data-driven forward-inverse problems for the variable coefficients Hirota (VCH) equation are discussed in this paper. The main idea is to use the improved physics-informed neural networks (IPINN) algorithm with neuron-wise locally adaptive activation function, slope recovery term and parameter regularization to recover the data-driven solitons and high-order soliton of the VCH equation with initial-boundary conditions, as well as the data-driven parameters discovery for VCH equation with unknown parameters under noise of different intensity. Numerical results are shown to demonstrate two facts: (i) data-driven soliton solutions of the VCH equation are successfully learned by adjusting the network layers, neurons, the original training data, spatiotemporal regions and other parameters of the IPINN algorithm; (ii) the prediction parameter can be trained stably and accurately by introducing a parameter regularization strategy with an appropriate weight coefficients into the IPINN algorithm. The results achieved in this work verify the effectiveness of the IPINN algorithm in solving the forward-inverse problems of the variable coefficients equation.