Solving the Biharmonic Equation with Coupled Physics-informed Neural Networks Based on Self-adaptive Loss Balancing
Journal: Journal of Higher Education Research DOI: 10.32629/jher.v7i1.4875
Abstract
Standard Physics-informed neural networks (PINNs) encounter notable challenges in solving this equation numerically, such as inadequate approximation accuracy, considerable computational expense, and difficulties in loss function optimization. To mitigate these limitations, we propose introducing an auxiliary variable to decompose the fourth-order biharmonic equation into a coupled system of two second-order partial differential equations. By separately approximating the original solution and the intermediate variable, this approach effectively alleviates the numerical errors associated with computing high-order derivatives while also reducing computational overhead. Furthermore, we incorporate an adaptive loss weighting scheme to dynamically balance the contributions of the PDE various PDE loss terms during training, thereby addressing the issue of loss imbalance inherent in multi-constrained learning. The effectiveness of the proposed method is validated through several numerical experiments.
Keywords
biharmonic equation, physics-informed neural networks, loss function, self-adaptive loss weighting
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[2] G. Chen, Z. L. Li, P. Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Adv. Comput. Math. 29 (2) (2008) 113-133.
[3] J. N. Flavin, Convexity considerations for the biharmonic equation in plane polars with applications to elasticity, Q. J. Mech. Appl. Math. 45 (4) (1992) 555-566.
[4] M. M. Gupta, A finite difference method for the biharmonic equation with applications to plane stress and plate bending problems, Comput. Struct. 9 (2) (1978) 133–137.
[5] N. J. Altiero, D. L. Sikarskie, A boundary integral method applied to plates of arbitrary plane form. Comput. Struct. 9 (2) (1978) 163–168.
[6] B. P. Lamichhane, A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems, J. Comput. Appl. Math. 235 (2011) 5188–5197.
[7] M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686-707.
[8] Z. Z. Xiang, W. Peng, X. Liu, W. Yao, Self-adaptive loss balanced physics-informed neural networks. Neurocomput. 496 (2022) 11–34.
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