Solving Parametric PDEs with Radial Basis Functions and Deep Neural Networks (2404.06834v2)
Abstract: We propose the POD-DNN, a novel algorithm leveraging deep neural networks (DNNs) along with radial basis functions (RBFs) in the context of the proper orthogonal decomposition (POD) reduced basis method (RBM), aimed at approximating the parametric mapping of parametric partial differential equations on irregular domains. The POD-DNN algorithm capitalizes on the low-dimensional characteristics of the solution manifold for parametric equations, alongside the inherent offline-online computational strategy of RBM and DNNs. In numerical experiments, POD-DNN demonstrates significantly accelerated computation speeds during the online phase. Compared to other algorithms that utilize RBF without integrating DNNs, POD-DNN substantially improves the computational speed in the online inference process. Furthermore, under reasonable assumptions, we have rigorously derived upper bounds on the complexity of approximating parametric mappings with POD-DNN, thereby providing a theoretical analysis of the algorithm's empirical performance.
- A reduced radial basis function method for partial differential equations on irregular domains. J. Sci. Comput., 66(1):67–90, 2016.
- Evaluation of proper orthogonal decomposition–based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput., 21(4):1419–1434, 1999.
- Data driven approximation of parametrized PDEs by reduced basis and neural networks. J. Comput. Phys., 416:109550, 2020.
- Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math., 23(3):317–330, 2005.
- The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat., 6(1):1–12, 2018.
- Numerical solution of the parametric diffusion equation by deep neural networks. J. Sci. Comput., 88(1):22, 2021.
- J. S. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. J. Comput. Phys., 363:55–78, 2018.
- Spectral Methods for Time-Dependent Problems, volume 21. Cambridge University Press, Cambridge, 2007.
- K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math., 90(1):117–148, 2001.
- A theoretical analysis of deep neural networks and parametric PDEs. Constr. Approx., 55(1):73–125, 2022.
- Solving parametric partial differential equations with deep rectified quadratic unit neural networks. J. Sci. Comput., 93(3):80, 2022.
- Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021.
- Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell., 3(3):218–229, 2021.
- Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng., 124(1):70–80, 2001.
- Reduced Basis Methods for Partial Differential Equations, volume 92. Springer, Cham, 2016.
- Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 378:686–707, 2019.
- A new look at proper orthogonal decomposition. SIAM J. Numer. Anal., 41(5):1893–1925, 2003.
- DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys., 375:1339–1364, 2018.
- Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification. J. Comput. Phys., 375:565–588, 2018.
- Holger Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2004.