»ã±¨±êÌâ (Title)£ºLift-and-Embed Learning Methods for Solving Scalar Hyperbolic Equations with Discontinuous Solutions

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»ã±¨ÈË (Speaker)£º Ëïç÷ ÖúÀí½ÌÊÚ£¨Í¬¼Ã´óѧ£©

»ã±¨¹¦·ò (Time)£º2024Äê11ÔÂ29ÈÕ(ÖÜÎå) 14:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

Ô¼ÇëÈË(Inviter)£ºÀîÐÂÏé

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»ã±¨ÌáÒª£ºUnlike traditional mesh-based discretizations of differential operators, machine learning methods, which exploit the automatic differentiation of neural networks to avoid dispersion and dissipation errors, have emerged as a compelling alternative in the numerical simulation of hyperbolic conservation laws. However, solutions to hyperbolic problems are often piecewise smooth, rendering the differential form invalid along discontinuity interfaces and limiting the efficacy of standard learning approaches. In this work, we propose lift-and-embed learning methods for solving scalar hyperbolic equations with discontinuous solutions, which consist of (i) embedding the Rankine-Hugoniot condition within a higher-dimensional space by including an augmented variable; (ii) utilizing neural networks to manage the increased dimensionality and to address both linear and nonlinear problems within a unified mesh-free learning framework; and (iii) projecting the trained network solution back onto the original plane to obtain the approximate solution. In addition, the location of discontinuity can be treated as unknown parameters and inferred concurrently with the training of network solution. With collocation points sampled on piecewise surfaces rather than fulfilling the lifted space, we conduct numerical experiments on various benchmark problems to demonstrate the capability of our methods in resolving discontinuous solutions without spurious smearing or oscillations.