Isogeometric neural networks: A new deep learning approach for solving parameterized partial differential equations

Abstract

We present a new physics-based machine learning approach for solving parameterized partial differential equations (PDEs) over generalized domains. Central to this method is the coupling of isogeometric representation with deep learning. This approach represents both the physical domain and the solution to a PDE as a linear combination of non-uniform rational B-spline (NURBS) basis functions and employs a deep feed-forward neural network to predict NURBS solution coefficients. Trained networks output the coefficients of NURBS to approximate PDE solutions for an entire range of physical and shape variables. Desirable aspects of the NURBS basis, such as its local support, refinement properties, and its capacity to represent freeform shapes compactly, are synergistically integrated with the ability of neural networks to learn hidden and non-linear mappings. Novel features of this isogeometric neural network (IGN) approach include its ability to perform physics-based learning over changing domains and efficient hierarchical learning. In contrast to classical PDE forward-solving techniques such as finite element analysis and isogeometric analysis, the IGN is able to learn PDE solutions for a range of parametric variables. We show empirically that no labeled data is required to obtain high-fidelity approximations to the solutions of parameterized PDEs. To demonstrate the efficacy of the IGN, 1D and 2D problems of various physics are addressed.