請用此 Handle URI 來引用此文件:
http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/52075
標題: | 透過有物理根據的神經網路求解高分子自洽場方程式: 正向與反向問題的應用 Physics-Informed Neural Networks for Solving Polymer Self-Consistent Field Equations: Applications to Forward and Inverse Problems |
作者: | Danny Yow-Chang Lin 林佑昌 |
指導教授: | 游琇伃(Hsiu-Yu Yu) |
關鍵字: | 深度學習,偏微分方程式,神經網路解偏微分方程式,自洽場理論,高分子流體, deep learning,partial differential equations,physics-informed neural networks,self-consistent field theory,polymeric fluids, |
出版年 : | 2020 |
學位: | 碩士 |
摘要: | 本研究中,我們採取了一個有物理根據的神經網路架構來解高分子自洽場理論之問題,傳統上此問題之解法已相當成熟,可以藉由pseudo-spectral method 配合迭代演算法來進行數值計算進而得到各高分子系統的結構。然而在此神經網路架構中,藉由損失函數把分別代表配分函數和自洽場的兩個神經網路結合在一起,讓每一次訓練可以同時求出配分函數和自洽場,省略了自洽場方程式的迭代過程。我們基於此架構,配合最近迅速發展之神經網路解偏微分方程的方法physics-informed neural networks (PINNs)來對其進行加強。本論文著重在以此架構重現前人已用傳統數值方法解出的高分子系統,並提出此神經網路架構之可能應用。 程式的編寫是使用Python配合深度學習框架PyTorch來進行,第一部分先藉由一些簡單的偏微分方程式來對PINNs進行測試,從直接解偏微分方程式、利用偏微分方程式之解來學習偏微分方程之參數、到最終探討此神經網路解偏微分方程時會遇到的反向傳播梯度不平均的問題。在此試著重現前人對於PINNs的研究並且同時測試程式的正確性和對此神經網路架構特性之了解。 接著我們把此方法應用到解高分子自洽場的問題上,我們進行了三種不同系統的計算,分別為Free Homopolymers、Tethered Homopolymers、和AB-block Copolymers的系統。 Free Homopolymers的系統是一個均質系統,主要拿來測試神經網路架構解自洽場方程式問題的可行性。Tethered Homopolymers在平行平板的系統在數值上為了要表示嫁接點,會需要使用到delta函數,這裡我們提出了一個改良的神經網路架構來處理delta函數在數值上不穩定的問題,結果上我們可以看到高分子嫁接在平板上的現象,但神經網路解出的答案無法呈現因為無溶劑狀態,高分子會因平板距離增加而被迫拉伸的現象。 AB-block Copolymers的系統在沒有對神經網路做特別的初始化之下,得到的結果都是均勻分布的無義解。我們藉由pseudo-spectral所解出的答案來先進行訓練,進而找到一組神經網路參數來作初始化,藉由此初始化我們可以成功解出不同高分子參數下所對應的結構。最後我們提出兩個應用,藉由此神經網路架構,我們可以設計想要的體積分率分布,來求出此分布所對應之自洽場;也可以反過來設計想要的自洽場,來解出相對應的體積分率和Flory-Huggins參數。 最終,本研究未來期望可以解決AB-block copolymers系統所遇到的均質無義解問題,同時也希望能藉由此神經網路之架構來解出三維AB-block Copolymers系統的結構。此外前段所提到的應用目前也只有拿已解出來的解當作測試資料,未來希望能對真實設計之體積分布和自洽場來進行測試。 In our research, we adopt a neural network-based architecture to solve the self-consistent field theory problems. Traditionally, the algorithms for solving SCFT problems are well-developed. One can use pseudo-spectral method with some iteration algorithm to numerically obtain the microstructure of polymeric fluids. Here, the NN-based architecture utilizes a loss function to couple the partition functions and self-consistent fields from their own network outputs. Within each iteration, partition functions and self-consistent fields are solved simultaneously. We further enhance this architecture by incorporating the techniques from the recently developing physics-informed neural networks (PINNs), which is a method to solve partial differential equations (PDEs) by neural networks. In this thesis, we tried to reproduce the well-known microstructures of polymeric fluids with PINNs framework. Then, taking advantage of this framework to propose two potential applications. The computations are conducted by Python codes with PyTorch. First, we take some simple PDE problems to test PINNs. From the forward problems, inverse problems and unbalanced back-propagated gradients problems, we try to reproduce the works of others in order to test our codes’ correctness and obtain a better understanding of PINNs. Then, we applied PINNs to the problem of SCFT. The focus is on three systems: free homopolymers, tethered homopolymer and AB-block copolymers. The system of free homopolymers is a homogeneous system. We take it to test the capability of this PINNs framework for SCFT problem. System of tethered homopolymers on a pair of parallel plates has to face the numerically unstable delta function due to the tethered point. We proposed an improved PINNs framework to cope with the delta function. In our results, we observe the tethered phenomena on the plates. However, PINNs fail to learn that the polymers are force to stretch when the gap size between plates increases. Without any specific initialization for PINNs, the system of AB-block copolymers suffers from trivial solution. We train our networks with the solution obtained from pseudo-spectral method to obtain a set of network parameters that can be taken as an initialization for FNNs. With this initialization, we successfully obtain the corresponding solution for different pairs of copolymers’ parameters. Moreover, we proposed two applications that take advantage of this framework. We can design a desired distribution of volume fraction to obtain its corresponding self-consistent fields. On the other hand, we can also design a desired self-consistent fields to find the corresponding volume fraction and Flory-Huggins parameter. Finally, our future goal is to resolve the problem of homogeneous solution for AB-block system. In the meantime, we would like to use this framework to further tackle the more complicated 3D system of AB-block copolymers. Lastly, the successes of our proposed applications are built upon using the known volume fraction and self-consistent fields as training data, we still need to test out their real capability on the self-designed data |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/52075 |
DOI: | 10.6342/NTU202002638 |
全文授權: | 有償授權 |
顯示於系所單位: | 化學工程學系 |
文件中的檔案:
檔案 | 大小 | 格式 | |
---|---|---|---|
U0001-0708202015110400.pdf 目前未授權公開取用 | 10.97 MB | Adobe PDF |
系統中的文件,除了特別指名其著作權條款之外,均受到著作權保護,並且保留所有的權利。