Spectral operator learning for parametric PDEs without data reliance
Junho Choi Taehyun Yun Namjung Kim Youngjoon Hong
In this paper, we introduce the Spectral Coefficient Learning via Operator
Network (SCLON), a novel operator learning-based approach for solving
parametric partial differential equations (PDEs) without the need for data
harnessing. The cornerstone of our method is the spectral methodology that
employs expansions using orthogonal functions, such as Fourier series and
Legendre polynomials, enabling accurate PDE solutions with fewer grid points.
By merging the merits of spectral methods - encompassing high accuracy,
efficiency, generalization, and the exact fulfillment of boundary conditions -
with the prowess of deep neural networks, SCLON offers a transformative
strategy. Our approach not only eliminates the need for paired input-output
training data, which typically requires extensive numerical computations, but
also effectively learns and predicts solutions of complex parametric PDEs,
ranging from singularly perturbed convection-diffusion equations to the
Navier-Stokes equations. The proposed framework demonstrates superior
performance compared to existing scientific machine learning techniques,
offering solutions for multiple instances of parametric PDEs without harnessing
data. The mathematical framework is robust and reliable, with a well-developed
loss function derived from the weak formulation, ensuring accurate
approximation of solutions while exactly satisfying boundary conditions. The
method's efficacy is further illustrated through its ability to accurately
predict intricate natural behaviors like the Kolmogorov flow and boundary
layers. In essence, our work pioneers a compelling avenue for parametric PDE
solutions, serving as a bridge between traditional numerical methodologies and
cutting-edge machine learning techniques in the realm of scientific
computation.