在地下水研究裡，達西定律（Darcy's Law）一直像是我們的基本信仰。教科書告訴我們：水流速度與壓力梯度成正比，這是一個線性、簡潔而優雅的關係，彷彿壓力一改變，流速就會瞬間回應。

但從 2017 年起，我開始懷疑這個「瞬間」是否真的存在。

當時我在熱傳領域看到「雙相延遲模型（Dual-Phase Lag Model）」，它描述雷射加熱金屬時，熱通量與溫度梯度之間會出現極微小的時間差。我開始思考：地下水流動是否也存在類似的慣性行為？

在微觀尺度下，水流穿越孔隙的路徑其實非常複雜。當我們用巨觀方程式描述它，許多微觀結構與孔隙阻力都被平均掉了。這些被忽略的細節，是否會在大尺度上表現成一種可觀測的「延遲」？

這就是後來我提出「延遲達西理論（Lagging Darcy Theory）」的出發點。核心概念很簡單：壓力梯度的產生與實際流動的發生，並非同一時間點。就像推動一個有慣性的重物，施力與位移之間總會有短暫時差。

而在 AI 快速發展後，我更確定這正是機器學習能發揮的場域。

人類擅長理解因果（causality）。我們知道水往低處流背後是重力與能量平衡，也知道延遲可能來自介質慣性；我們可以建立物理框架，描述世界「應該」如何運作。

AI 則擅長發現關聯（correlation）。真實地層有隨機裂隙、未知邊界與非線性效應，很多現象超出手寫公式的表達能力。過去我們把它們當誤差，現在我更傾向把這些複雜殘差交給 AI 學習。

在我的實驗室，我們不做二選一。我們以物理模型建立骨幹，確保推論符合守恆定律；再用 AI 學習物理模型尚未解釋的殘差結構。

這就是我理解的人機協作：人類定義物理邊界，AI 補足真實世界的細節。AI 不是取代理論，而是讓我們看見過去公式裡看不見的那個時間差。

In groundwater science, Darcy's Law has long been our foundational belief. Textbooks teach us that flow velocity is proportional to pressure gradient: a clean, elegant linear relation, as if flow responds instantly once pressure changes.

Since 2017, however, I have questioned that assumption of instantaneity.

At the time, I was inspired by the Dual-Phase Lag model in heat transfer, which explains how heat flux and temperature gradient can be slightly out of phase under ultrafast laser heating. I then asked: could groundwater flow exhibit a similar lag behavior?

At the pore scale, fluid motion is highly complex. When we describe it with a macroscopic equation, many microstructural effects and pore-level resistance are averaged out. Those neglected details may reappear, at larger scales, as an observable delay.

This led to what I later called the Lagging Darcy Theory. The central idea is straightforward: the generation of pressure gradient and the realization of flow do not occur at exactly the same time. Like pushing a heavy object with inertia, force and movement are separated by a small time gap.

With the rise of AI, I became convinced that this is exactly where machine learning can contribute.

Humans are strong at causality. We understand why water flows downhill and why lag may emerge from medium inertia. We can build physically meaningful structures that describe how the system should behave.

AI is strong at correlation. Real geological settings contain random fractures, unknown boundaries, and nonlinear effects that exceed the expressive limits of handwritten formulas. What we once dismissed as error can now be learned as structured residual information.

In my lab, we do not force a binary choice. We use physics to define the backbone and preserve conservation laws, then use AI to learn the residual patterns that the physics model cannot fully capture.

This is my view of human-AI collaboration: humans define physical boundaries, while AI fills in the details of the real world. AI does not replace theory; it reveals the time lag that classical formulas could not fully see.