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A class of physics-informed spatio-temporal models has recently been proposedfor modeling spatio-temporal processes governed by advection-diffusionequations. The central idea is to approximate the process by a truncatedFourier series and let the governing physics determine the dynamics of thespectral coefficients. However, because many spatio-temporal processes in realapplications are non-periodic with boundary discontinuities, the well-knownGibbs phenomenon and ripple artifact almost always exist in the outputsgenerated by such models due to truncation of the Fourier series. Hence, thekey contribution of this paper is to propose a physics-informed spatio-temporalmodeling approach that significantly suppresses the Gibbs phenomenon whenmodeling spatio-temporal advection-diffusion processes. The proposed approachstarts with a data flipping procedure for the process respectively along thehorizontal and vertical directions (as if we were unfolding a piece of paperthat has been folded twice along the two directions). Because the flippedprocess becomes spatially periodic and has a complete waveform without anyboundary discontinuities, the Gibbs phenomenon disappears even if the Fourierseries is truncated. Then, for the flipped process and given the PartialDifferential Equation (PDE) that governs the process, this paper extends anexisting PDE-based spatio-temporal model by obtaining the new temporal dynamicsof the spectral coefficients, while maintaining the physical interpretation ofthe flipped process. Numerical investigations based on a real dataset have beenperformed to demonstrate the advantages of the proposed approach. It is foundthat the proposed approach effectively suppresses the Gibbs Phenomenon andsignificantly reduces the ripple artifact in modeling spatio-temporaladvection-diffusion processes. Computer code is available on GitHub.

Gibbs Phenomenon Suppression in PDE-Based Statistical Spatio-Temporal Models