feat: 后向自动微分

include/dual.hpp

@@ -3,6 +3,7 @@
 #include <iostream>
 #include <vector>
 #include <cmath>
+#include "types/common.hpp"
 #include "types/dual.hpp"
 
 // 前向自动微分的运算部分，两个deriv相乘为0
@@ -62,4 +63,123 @@ namespace forwardad{
     inline Dual atan(const Dual& x) {
         return Dual(std::atan(x.value), x.deriv / (1 + x.value * x.value));
     }
-}// namespace forwardad
+}// namespace forwardad
+
+namespace backwardad {
+    inline Node operator+(const Node& a, const Node& b) {
+        Node out(a.value() + b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient()); // da = dout * 1
+            b.add_gradient(out.gradient()); // db = dout * 1
+        };
+        return out;
+    }
+
+    inline Node operator-(const Node& a, const Node& b) {
+        Node out(a.value() - b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient()); // da = dout * 1
+            b.add_gradient(-out.gradient()); // db = dout * (-1)
+        };
+        return out;
+    }
+
+    inline Node operator*(const Node& a, const Node& b) {
+        Node out(a.value() * b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient() * b.value()); // da = dout * b
+            b.add_gradient(out.gradient() * a.value()); // db = dout * a
+        };
+        return out;
+    }
+
+    inline Node operator/(const Node& a, const Node& b) {
+        Node out(a.value() / b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient() / b.value()); // da = dout * (1/b)
+            b.add_gradient(-out.gradient() * a.value() / (b.value() * b.value())); // db = dout * (-a/b^2)
+        };
+        return out;
+    }
+
+    // 下面的函数需要用到链式法则(使用泰勒展开后保留一次项)
+    // 三角函数
+    inline Node sin(const Node& x) {
+        Node out(std::sin(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() * std::cos(x.value())); // dx = dout * cos(x)
+        };
+        return out;
+    }
+
+    inline Node cos(const Node& x) {
+        Node out(std::cos(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(-out.gradient() * std::sin(x.value())); // dx = dout * (-sin(x))
+        };
+        return out;
+    }
+
+    // 指数和对数
+    inline Node exp(const Node& x) {
+        Node out(std::exp(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() * out.value()); // dx = dout * exp(x) = dout * out.value
+        };
+        return out;
+    }
+
+    inline Node log(const Node& x) {
+        Node out(std::log(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / x.value()); // dx = dout * (1/x)
+        };
+        return out;
+    }
+
+    inline Node pow(const Node& x, double n) {
+        Node out(std::pow(x.value(), n));
+        out.inputs() = {x};
+        out.backward() = [out, x, n]() {
+            x.add_gradient(out.gradient() * n * std::pow(x.value(), n - 1)); // dx = dout * (n * x^(n-1))
+        };
+        return out;
+    }
+
+    // 反三角函数
+    inline Node asin(const Node& x) {
+        Node out(std::asin(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / std::sqrt(1 - x.value() * x.value())); // dx = dout * (1/sqrt(1-x^2))
+        };
+        return out;
+    }
+
+    inline Node acos(const Node& x) {
+        Node out(std::acos(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(-out.gradient() / std::sqrt(1 - x.value() * x.value())); // dx = dout * (-1/sqrt(1-x^2))
+        };
+        return out;
+    }
+
+    inline Node atan(const Node& x) {
+        Node out(std::atan(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / (1 + x.value() * x.value())); // dx = dout * (1/(1+x^2))
+        };
+        return out;
+    }
+
+}