使用計算機來解決偏微分方程，我們通常會使用數值方法來解，例如有限差分法、有限元素法，或是使用機器學習來解決這些問題，通常是以逼近函數的方式來進行求解的過程。而近年來又發展了算子學習，我們可以直接學習方程式本身，進而提升學習結果的泛化能力。為了瞭解算子學習與非算子學習之間的差異，將會使用傅立葉神經算子和物理訊息神經網路對同一問題進行訓練，並比較其學習成本以及結果誤差。實驗表明，算子學習在學習成本上較高，但訓練後的模型較有泛化能力，對於特定的問題會有極佳的成效。

To solve partial differential equations using computers, we typically employ numerical methods such as finite difference methods, finite element methods, or machine learning. This often involves the process of approximating functions to attain a solution. In recent years, operator learning has also emerged, enabling direct learning of the equations, thereby enhancing the generalization ability of the learning results. To understand the differences between operator learning and non-operator learning, we will employ Fourier Neural Operator and Physics-informed Neural Networks to train on the same problem, subsequently comparing their learning costs and error outcomes. Experiments indicate that operator learning incurs higher learning costs, but the trained model exhibits superior generalization ability, leading to excellent performance on specific problems.

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