Implicit Legendre Representation for Physics-informed Neural Network

Physics-informed neural networks (PINNs) commonly employ Fourier-based representations to more accurately approximate high-frequency components in solutions to partial differential equations (PDEs). In this work, we challenge the need for spectral priors in representation design and propose Legendre representation networks (LERENs), which integrate learnable Legendre expansions into multilayer perceptrons (MLPs). We evaluate LERENs on five challenging PDE problems involving high-frequency components, complex solution structures, and high-dimensional domains. Experimental results demonstrate that LERENs outperform state-of-the-art Fourier-based representation methods on the evaluated benchmarks, establishing Legendre-based representations as a competitive alternative to Fourier-based representations in PINNs. The proposed representation paradigm may extend to other bases used in spectral methods, allowing their favorable mathematical properties to inform the design of neural representations.