分佈式參數系統 (DPS) 的模型對於物理、生物以及工程系統中的設計和控制極為重要。這種系統可以用一個邊界條件固定但操作值函數和初始函數可變的偏微分方程表示：y\left(z,t\right)=f\left(z,t,y\left(z,0\right),u\left(z,t\right)\right)，此方程亦可稱為輸入函數為操作值函數和初始函數的泛函。
　　本研究中，我們建立DeepONet神經網路，並基於其結構發展出另外兩種以正交多項式逼近的結構，對分佈式參數變溫系統進行建模，並以不同訓練數據進行動態預測以及以後者模型進行參數預測。
　　在動態預測中，我們由DeepONet神經網路的訓練結果可以發現，當其為「全深度神經網路」時，模型訓練結果不甚理想，而「主幹及分支為深度神經網路」的預測結果隨著訓練樣本數的增加而有所提升，然而「僅分支網路為深度神經網路」的表現卻能比前者來的更好，給予500組訓練樣本數，其預測R2中位數甚至能達到0.99；而參數預測中，我們發現改變不同損失值的權重會影響參數預測的準確性，然而當權重失衡時會導致預測準確急遽下降，而此模型在參數預測最好的情況也能達到誤差值0.09以下。
　　總而言之，藉由上述實驗，我們驗證深度算子神經網路(DeepONet)對於分布式參數系統能有很好的建模能力，且正交函數替代主幹網路亦能對模型訓練結果及參數預測有正面的提升。

Model representation of a distributed parameter system (DPS) is extremely important for design and control in process system engineering. Such system can be represented by a partial differential equation with fixed boundary condition, but variable activation function and initial condition: y\left(z,t\right)=f\left(z,t,y\left(z,0\right),u\left(z,t\right)\right). Such a system can be referred to as an operator on functions y\left(z,0\right) and u\left(z,t\right); or a functional of z, and t.
It has been well-known in machine learning that an artificial neural network is an universal approximator. Lu et al 2021 demonstrated that artificial neural network can also be an universal “functional” or “operator” approximator and that the training can be much facilitated by dot product of a “branch-net” and a “trunk-net”. The “branch-net” received input from initial conditions and the input to the “trunk-net” is the point of interests in space-time. A neural network representation of a DPS, known as DeepOnet can be developed by a data-driven approach, or a first principle approach, or a hybrid approach as propose in the physics informed neural network (PINN).
　　We found that DeepOnet is neural network implementation of PDE solution by orthogonal approximation of the functional/operator. Hence DeepONet can actually be divided into five subunits: space-trunk-net, time-trunk-net, initial condition-trunk-net and activation trunk-net and a branch-net. While all of them can be left as a free form and trained for a specific problem, each of these trunknets can be pretrained using orthogonal polynomials for a specific space-time domain. Only the parameters of the branch-net must be trained for the specific problem.
　　The simplified approach was demonstrated the forward solution of a heat transfer problem studied by Gay and Ray 1995, and showed that using orthogonal DeepONet, identification and solution of the DPS can be much simplified.

摘要 2
Abstract 3
致謝 4
目錄 5
圖目錄 7
第一章 緒論 9
一.1 研究背景 9
一.2 文獻回顧 10
一. 2.1 神經網路與深度學習 10
一. 2.1.1 神經網路架構 10
一. 2.1.2 深度學習 11
一. 2.2 物理結構導入神經網路模型 13
一. 2-2.1 前向求解器 14
一. 2.2.2 逆向求解器 14
一. 2.3 深度算子神經網路模型 15
一. 2.3.1 深度算子神經網路模型結構 16
一. 2.3.2 深度算子神經網路模型與傳統模型比較 17
一. 3 研究動機與目標 17
第二章 深度算子神經網路與正交函數逼近法結構 19
二.1 正交多項式在解PDE 的應用 19
二.2 DeepONet-1 (全深度神經網路) 21
二.3 DeepONet-2 (主幹及分支為深度神經網路) 22
二.4 DeepONet -3(僅分支網路為深度神經網路) 結構 24
第三章 案例介紹與數據產生及模型訓練方法 26
三.1 變溫系統 26
三.2 數據生成 26
三.3訓練方法 28
第四章 模型訓練結果 31
四.1 DeepONet-1模型動態預測 31
四.2 DeepONet-2模型動態預測 31
四.3 DeepONet-3 模型動態預測 32
四.4 三種DeepONet 動態預測比較 33
四.5 DeepOnet-3參數預測 36
第五章 結論 37
參考文獻 38

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