POSITION
当前位置Current position
一组参数决定球在曲面上的位置。A parameter pair locates the ball.
01 · 站在山坡上01 · Stand on a hillside
不用看整座山，只看脚下的一小块，怎样知道哪边上升最快？Without seeing the whole mountain, can the patch under your feet reveal the fastest way up?
02把球放到损失曲面，读取局部箭头Place the Ball and Read the Local Arrows
唯一变量：球的位置One variable: ball positionθ1
2.35
参数 1parameter 1
θ2
1.80
参数 2parameter 2
‖∇L‖
—
局部陡峭度local steepness
正在加载曲面；若提示持续显示，则需要 WebGL /
three.js。Loading surface; if this remains visible, WebGL or three.js is unavailable.
粉红 = +∇L · 青色 = −∇Lpink = +∇L · cyan = −∇L
双击放球 · 拖动旋转double-click to place · drag to orbit
梯度：最陡上升gradient: steepest ascent
反梯度：下降方向negative gradient: descent
绿环：曲面低谷green ring: basin floor
03梯度不是一个神秘箭头A Gradient Is Not a Mysterious Arrow
position · partials · vector · directionPARTIALS
分方向变化率Per-axis rates
分别问每个参数轻微增加会怎样。Ask what a small increase on each axis does.
VECTOR
合成向量Combined vector
各方向变化率组成 ∇L。Axis-wise rates form ∇L.
LOCAL
只保证局部Local only
换一个位置，箭头会重新计算。Move elsewhere and the arrow changes.
04最小机制：两个偏导组成一个方向Minimal Mechanism: Two Partial Derivatives Form One Direction
∇L = [∂L/∂θ1, ∂L/∂θ2]∇L(θ) = [∂L/∂θ1, ∂L/∂θ2]
箭头方向给出最陡上升；箭头长度 ‖∇L‖
表示局部有多陡。下降要取负号。Direction gives steepest ascent; length ‖∇L‖ gives local steepness. Descent
uses the negative.
移动球Move ball参数位置改变parameters change
重算局部斜率Recompute local slopes两个偏导变化partials change
箭头转向缩放Arrow turns and scales显示最陡方向shows steepest direction
05边界实验：在谷底寻找“大梯度”Boundary Test: Demand a Large Gradient at the Bottom
主动制造失败Create a failure把球放到最低附近Place the ball near the minimum
损失仍是一个正数，但局部几乎平坦，梯度接近零。这直接证明“梯度 ≠ 误差”。Loss remains positive while the surface is locally flat and gradient nears zero. This directly shows gradient is not error.
尚未执行。Not run yet.
✗ 梯度就是损失值。Gradient is the loss value.
→ ✓ 梯度描述损失对参数的局部变化率。Gradient describes local loss change with parameters.
✗ 沿梯度走会降低损失。Following the gradient lowers loss.
→ ✓ 梯度指上坡，下降要沿 −∇L。Gradient points uphill; descent follows −∇L.
✗ 一个梯度方向适用于整座曲面。One gradient direction works everywhere.
→ ✓ 它只属于当前参数位置。It belongs only to the current parameter position.
06 · 一句话带走06 · One line to keep