A Kernel Approach for PDE Discovery and Operator Learning

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its competitive performance

Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment

A realistic interpretation(REIN) of wave function in quantum mechanics is briefly presented in this work. In REIN, the wave function of a microscopic object is just its real existence rather than a mere mathematical description. Quantum object can exist in disjoint regions of space which just as the wave function distributes, travels at a finite speed, and collapses instantly upon a measurement. The single photon interference in a Mach-Zehnder interferometer is analyzed using REIN. In particular, we proposed and experimentally implemented a generalized delayed-choice experiment, the encounter-delayed-choice(EDC) experiment, in which the second beam splitter is inserted at the encounter of the two sub-waves from the two arms. In the EDC experiment, the front parts of wave functions before the beam splitter insertion do not interfere and show the particle nature, and the back parts of the wave functions will interfere and show a wave nature. The predicted phenomenon is clearly demonstrated in the experiment, and supports the REIN idea.Comment: 7 pages 4 figure