Article: Simulation modelling inverse problems of partial differential equations with physics-informed neural networks Journal: International Journal of Simulation and Process Modelling (IJSPM) 2026 Vol.23 No.6 pp.13 - 29 Abstract: Inverse problems of partial differential equations are inherently ill-posed, presenting a fundamental challenge in simulation modelling as they yield unstable solutions from sparse and noisy measurements. To address this, we introduce a regularised physics-informed neural network framework. Its key methodological innovation is the systematic integration of explicit prior constraints - such as total variation and sparsity regularisation - with an adaptive weighting scheme, thereby bridging the theoretical rigour of classical inverse problems with the flexibility of deep learning. The framework is qualitatively validated on several benchmark problems, including nonlinear transport and porous media flow, where it consistently demonstrates superior performance over existing baselines in terms of accuracy, robustness to noise, and structural fidelity. This work advances simulation modelling by offering a stable, mesh-free, and prior-aware tool, with potential impact on inverse problem solving in geophysics, medical imaging, and engineering design. Inderscience Publishers - linking academia, business and industry through research

Title: Simulation modelling inverse problems of partial differential equations with physics-informed neural networks

Authors: Min Guo

Addresses: Basic Teaching Department, Shanghai Zhongqiao Vocational and Technical University, Shanghai, 201514, China

Abstract: Inverse problems of partial differential equations are inherently ill-posed, presenting a fundamental challenge in simulation modelling as they yield unstable solutions from sparse and noisy measurements. To address this, we introduce a regularised physics-informed neural network framework. Its key methodological innovation is the systematic integration of explicit prior constraints - such as total variation and sparsity regularisation - with an adaptive weighting scheme, thereby bridging the theoretical rigour of classical inverse problems with the flexibility of deep learning. The framework is qualitatively validated on several benchmark problems, including nonlinear transport and porous media flow, where it consistently demonstrates superior performance over existing baselines in terms of accuracy, robustness to noise, and structural fidelity. This work advances simulation modelling by offering a stable, mesh-free, and prior-aware tool, with potential impact on inverse problem solving in geophysics, medical imaging, and engineering design.

Keywords: regularised physics-informed neural network; partial differential equations; inverse problems; simulation modelling; adaptive weighting; regularisation techniques.

DOI: 10.1504/IJSPM.2026.153268

International Journal of Simulation and Process Modelling, 2026 Vol.23 No.6, pp.13 - 29

Received: 17 Dec 2025
Accepted: 13 Mar 2026
Published online: 29 Apr 2026 *

Title: Simulation modelling inverse problems of partial differential equations with physics-informed neural networks

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