算一算，我研究解析解（Analytical Solution）已經超過十一年。從博士班到博後，無數夜晚都在推導那些可能包含無窮級數的方程式。

學生常問我：「老師，現在數值模式這麼強，甚至 AI 也能快速產生結果，為什麼還要花這麼多力氣推導解析解？」

這是一個好問題。對我來說，解析解不只是答案，而是一把「尺」。

解析解是在嚴格假設下得到的精確數學解，例如均質地層、無限邊界等。它代表物理上的理想狀態。當然，我們都知道真實地下環境絕不理想，存在斷層、污染與各種非均質介質。

但正因為現實複雜，我們更需要這把尺。沒有物理基準，面對混亂數據時，你甚至無法判斷模型偏離了多少。

近年我的研究逐漸轉向「物理導向機器學習（Physics-Guided AI）」，本質上是把過去十多年的物理建模能力，和新一代資料方法整合。

我們不再嘗試用單一完美公式強行解釋所有現象，也不盲信黑盒子的預測。我的做法是先用解析解建立基準（benchmark），定義在理想條件下，地下水流與熱傳應該呈現的行為。

接著把真實觀測資料疊上去，兩者之間必然存在落差，也就是殘差（residual）。

過去很多人把殘差視為雜訊或測量誤差。但在機器學習框架下，我們看到殘差其實含有高價值資訊：它可能反映地層非均質性，也可能暗示尚未被揭示的物理機制。

所以在我的實驗室，數學與 AI 不是對立關係。解析解是導航員，AI 是探險車；前者確保方向正確，後者幫助我們穿越現地的崎嶇地形。

這也是到了 2026 年，我仍堅持推導公式的原因。唯有深刻理解物理本質，才有能力駕馭 AI，而不是被資料牽著走。

I have been working on analytical solutions for more than eleven years. From my PhD to postdoctoral training, I spent countless nights deriving equations, sometimes with infinite-series structures.

Students often ask: "With powerful numerical models and AI tools now available, why still invest so much effort in analytical derivation?"

For me, an analytical solution is not just an answer; it is a ruler.

Analytical solutions are exact mathematical results under strict assumptions, such as homogeneous formations and idealized boundaries. They represent the physics in a controlled, ideal state. Real subsurface environments are never that clean: faults, contamination, and heterogeneity are everywhere.

Precisely because reality is messy, we need that ruler. Without a physical reference, once you face complex data, you cannot even tell how far your model has drifted.

In recent years, my work has moved toward Physics-Guided AI. In essence, this is a synthesis: combining long-developed physical modeling capability with modern data-driven methods.

We neither force reality into one perfect formula nor blindly trust black-box predictions. My workflow starts with an analytical benchmark that defines how groundwater flow and heat transport should behave under ideal conditions.

Then we overlay real observations. The gap between benchmark and reality is inevitable: this is the residual.

Traditionally, residuals were treated as noise or measurement error. With machine learning, however, we now see residuals as information-rich signals. They may encode geological heterogeneity or indicate physical mechanisms not yet identified.

So in my lab, mathematics and AI are not rivals. Analytical solutions are our navigator; AI is our exploration vehicle. One keeps us on the right direction, the other helps us traverse rough terrain.

That is why, even in 2026, I still insist on deriving formulas. Only by understanding physical fundamentals can we truly control AI, instead of being controlled by data.