在這篇論文中，我們提供一種改良版的守恆物理性神經網路來建構非線性守恆方程的黎曼問題解。這些守恆方程可以是寫成守恆或非守恆型式。如果是守恆型式，方程中的通量包含了一個不連續的擾動量。我們提供廣義的巴克利-萊弗里特方程來展示我們的研究結果。這個方程式代表了多孔介質流體在不連續孔隙率介質中的活動行為。藉由引進一個新的未知量，這個巴克利-萊弗里特方程可以被轉換成一個非線性共振守恆系統且帶有非連續的擾動項。我們用守恆物理性神經網路針對守恆或非守恆巴克利-萊弗里特系統在非臨界或臨界黎曼初始值的案例中造出弱解。針對此方程系統的性質，我們把時空間分割成二個區間，第一區間解一個二維系統，第二區間解一個單一非凸的守恆方程的黎曼問題，且針對不同區間給定不同的損失函數。當黎曼初始值靠近臨界值時，我們引進了重新縮放尺度技巧，適度的選取縮放尺度的參數使得弱解沒有振盪的行為。所以由守恆物理性神經網路法所得的解可以完美的匹配由傳統偏微分方程所得到的解。我們也比較了此方法和WENO 5數值方法的優缺點，我們發現此方法可以用在非守恆型式方程，但WENO 5方法則在這類型方程有一些困難度。最後，我們也研究了把元學習方法加入守恆物理性神經網路法的可行性。;In this dissertation, a modified version of conservative Physics-informed Neural Networks (cPINN) is provided to construct the solutions of Riemann problem for the hyperbolic scalar conservation laws in non-conservative form and scalar conservation laws with discontinuous perturbation in the flux. To demonstrate the results, we use the model of generalized Buckley-Leverett equation (GBL equation) with discontinuous porosity in porous media. By inventing a new unknown, the GBL equation is transformed into a two-by-two resonant hyperbolic conservation laws in conservative form. We experiment with our idea by using a cPINN algorithm to solve the GBL equation in both conservative and non-conservative forms, as well as the cases of critical and non-critical states. This method provides a combination of two different neural networks and corresponding loss functions, one is for the two-by-two resonant hyperbolic system, and the other is for the scalar conservation law with a discontinuous perturbation term in the non-convex flux. The technique of re-scaling to the unknowns is adopted to avoid the oscillation or inaccurate speed of the Riemann solutions in the cases of critical Riemann data. The solutions constructed by the modified cPINN match the exact solutions constructed by the theoretical analysis for hyperbolic conservation laws. Finally, we compare the performance of the modified cPINN with numerical method WENO5. Whereas WENO5 struggles with the approximate solutions for the Riemann problems of GBL equation in non-conservative form, cPINN works admirably. Moreover, the integration of meta-learning into the initialization of cPINN for the re-scaled GBL equation is being explored as an additional step forward.
