IGA-ODIL: Fast Physics-Informed Learning with B-Spline Representations

Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss-Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second-order optimization methods. In this work, we introduce IGA-ODIL (Optimization of DIscrete Loss with data representation following from IsoGeometric Analysis). Instead of neural-network parameterizations of PINNs, the unknown solution is represented by smooth B-spline basis functions, leading to structured, sparse Jacobians and efficient second-order Gauss–Newton optimization. For the nonlinear inverse problem, we use a damped Gauss–Newton (Levenberg–Marquardt) variant to improve robustness. In the considered implementations, the resulting sparse second-order optimization achieves two to three orders of magnitude lower wall-clock time than the corresponding first-order PINN and CRVPINN implementations. The resulting systems inherit locality, sparsity, and approximation-theoretic properties of classical finite element and isogeometric methods while preserving the residual-learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection-dominated advection–diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen–Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders-of-magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.