kmath-core/src/commonMain/kotlin/scientifik/kmath/operations/AutoDiff.kt

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+package scientifik.kmath.operations
+
+import kotlin.math.pow
+import kotlin.math.sqrt
+
+/*
+ * Implementation of backward-mode automatic differentiation.
+ * Initial gist by Roman Elizarov: https://gist.github.com/elizarov/1ad3a8583e88cb6ea7a0ad09bb591d3d
+ */
+
+/**
+ * Differentiable variable with value and derivative of differentiation ([deriv]) result
+ * with respect to this variable.
+ */
+data class ValueWithDeriv(var x: Double, var d: Double = 0.0) {
+    constructor(x: Number) : this(x.toDouble())
+}
+
+/**
+ * Runs differentiation and establishes [Field<ValueWithDeriv>] context inside the block of code.
+ *
+ * Example:
+ * ```
+ * val x = ValueWithDeriv(2) // define variable(s) and their values
+ * val y = deriv { sqr(x) + 5 * x + 3 } // write formulate in deriv context
+ * assertEquals(17.0, y.x) // the value of result (y)
+ * assertEquals(9.0, x.d)  // dy/dx
+ * ```
+ */
+fun deriv(body: AutoDiffField.() -> ValueWithDeriv): ValueWithDeriv =
+    ValueWithDerivField().run {
+        val result = body()
+        result.d = 1.0 // computing derivative w.r.t result
+        runBackwardPass()
+        result
+    }
+
+
+abstract class AutoDiffField : Field<ValueWithDeriv> {
+    /**
+     * Performs update of derivative after the rest of the formula in the back-pass.
+     *
+     * For example, implementation of `sin` function is:
+     *
+     * ```
+     * fun AD.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sin(x.x)) { z -> // call derive with function result
+     *     x.d += z.d * cos(x.x) // update derivative using chain rule and derivative of the function
+     * }
+     * ```
+     */
+    abstract fun <R> derive(value: R, block: (R) -> Unit): R
+
+    // Overloads for Double constants
+
+    operator fun Number.plus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() + that.x)) { z ->
+        that.d += z.d
+    }
+
+    operator fun ValueWithDeriv.plus(b: Number): ValueWithDeriv = b.plus(this)
+
+    operator fun Number.minus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() - that.x)) { z ->
+        that.d -= z.d
+    }
+
+    operator fun ValueWithDeriv.minus(that: Number): ValueWithDeriv = derive(ValueWithDeriv(this.x - that.toDouble())) { z ->
+        this.d += z.d
+    }
+
+    override operator fun Number.times(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() * that.x)) { z ->
+        that.d += z.d * this.toDouble()
+    }
+
+    override operator fun ValueWithDeriv.times(b: Number): ValueWithDeriv = b.times(this)
+
+    override operator fun Number.div(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() / that.x)) { z ->
+        that.d -= z.d * this.toDouble() / (that.x * that.x)
+    }
+
+    override operator fun ValueWithDeriv.div(that: Number): ValueWithDeriv = derive(ValueWithDeriv(this.x / that.toDouble())) { z ->
+        this.d += z.d / that.toDouble()
+    }
+}
+
+/**
+ * Automatic Differentiation context class.
+ */
+private class ValueWithDerivField : AutoDiffField() {
+
+    // this stack contains pairs of blocks and values to apply them to
+    private var stack = arrayOfNulls<Any?>(8)
+    private var sp = 0
+
+
+    @Suppress("UNCHECKED_CAST")
+    override fun <R> derive(value: R, block: (R) -> Unit): R {
+        // save block to stack for backward pass
+        if (sp >= stack.size) stack = stack.copyOf(stack.size * 2)
+        stack[sp++] = block
+        stack[sp++] = value
+        return value
+    }
+
+    @Suppress("UNCHECKED_CAST")
+    fun runBackwardPass() {
+        while (sp > 0) {
+            val value = stack[--sp]
+            val block = stack[--sp] as (Any?) -> Unit
+            block(value)
+        }
+    }
+
+    // Basic math (+, -, *, /)
+
+
+    override fun add(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
+        derive(ValueWithDeriv(a.x + b.x)) { z ->
+            a.d += z.d
+            b.d += z.d
+        }
+
+    override fun multiply(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
+        derive(ValueWithDeriv(a.x * b.x)) { z ->
+            a.d += z.d * b.x
+            b.d += z.d * a.x
+        }
+
+    override fun divide(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
+        derive(ValueWithDeriv(a.x / b.x)) { z ->
+            a.d += z.d / b.x
+            b.d -= z.d * a.x / (b.x * b.x)
+        }
+
+    override fun multiply(a: ValueWithDeriv, k: Number): ValueWithDeriv =
+        derive(ValueWithDeriv(k.toDouble() * a.x)) { z ->
+            a.d += z.d * k.toDouble()
+        }
+
+    override val zero: ValueWithDeriv get() = ValueWithDeriv(0.0, 0.0)
+    override val one: ValueWithDeriv get() = ValueWithDeriv(1.0, 0.0)
+}
+
+// Extensions for differentiation of various basic mathematical functions
+
+// x ^ 2
+fun AutoDiffField.sqr(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(x.x * x.x)) { z ->
+    x.d += z.d * 2 * x.x
+}
+
+// x ^ 1/2
+fun AutoDiffField.sqrt(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sqrt(x.x))) { z ->
+    x.d += z.d * 0.5 / z.x
+}
+
+// x ^ y (const)
+fun AutoDiffField.pow(x: ValueWithDeriv, y: Double): ValueWithDeriv = derive(ValueWithDeriv(x.x.pow(y))) { z ->
+    x.d += z.d * y * x.x.pow(y - 1)
+}
+
+fun AutoDiffField.pow(x: ValueWithDeriv, y: Int): ValueWithDeriv = pow(x, y.toDouble())
+
+// exp(x)
+fun AutoDiffField.exp(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.exp(x.x))) { z ->
+    x.d += z.d * z.x
+}
+
+// ln(x)
+fun AutoDiffField.ln(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.ln(x.x))) { z ->
+    x.d += z.d / x.x
+}
+
+// x ^ y (any)
+fun AutoDiffField.pow(x: ValueWithDeriv, y: ValueWithDeriv): ValueWithDeriv = exp(y * ln(x))
+
+// sin(x)
+fun AutoDiffField.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.sin(x.x))) { z ->
+    x.d += z.d * kotlin.math.cos(x.x)
+}
+
+// cos(x)
+fun AutoDiffField.cos(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.cos(x.x))) { z ->
+    x.d -= z.d * kotlin.math.sin(x.x)
+}

kmath-core/src/commonTest/kotlin/scientifik/kmath/operations/AutoDiffTest.kt

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+package scientifik.kmath.operations
+
+import kotlin.math.PI
+import kotlin.test.Test
+import kotlin.test.assertEquals
+
+class AutoDiffTest {
+    @Test
+    fun testPlusX2() {
+        val x = ValueWithDeriv(3) // diff w.r.t this x at 3
+        val y = deriv { x + x }
+        assertEquals(6.0, y.x) //    y  = x + x = 6
+        assertEquals(2.0, x.d) // dy/dx = 2
+    }
+
+    @Test
+    fun testPlus() {
+        // two variables
+        val x = ValueWithDeriv(2)
+        val y = ValueWithDeriv(3)
+        val z = deriv { x + y }
+        assertEquals(5.0, z.x) //    z  = x + y = 5
+        assertEquals(1.0, x.d) // dz/dx = 1
+        assertEquals(1.0, y.d) // dz/dy = 1
+    }
+
+    @Test
+    fun testMinus() {
+        // two variables
+        val x = ValueWithDeriv(7)
+        val y = ValueWithDeriv(3)
+        val z = deriv { x - y }
+        assertEquals(4.0, z.x)  //    z  = x - y = 4
+        assertEquals(1.0, x.d)  // dz/dx = 1
+        assertEquals(-1.0, y.d) // dz/dy = -1
+    }
+
+    @Test
+    fun testMulX2() {
+        val x = ValueWithDeriv(3) // diff w.r.t this x at 3
+        val y = deriv { x * x }
+        assertEquals(9.0, y.x) //    y  = x * x = 9
+        assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
+    }
+
+    @Test
+    fun testSqr() {
+        val x = ValueWithDeriv(3)
+        val y = deriv { sqr(x) }
+        assertEquals(9.0, y.x) //    y  = x ^ 2 = 9
+        assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
+    }
+
+    @Test
+    fun testSqrSqr() {
+        val x = ValueWithDeriv(2)
+        val y = deriv { sqr(sqr(x)) }
+        assertEquals(16.0, y.x) //     y = x ^ 4   = 16
+        assertEquals(32.0, x.d) // dy/dx = 4 * x^3 = 32
+    }
+
+    @Test
+    fun testX3() {
+        val x = ValueWithDeriv(2) // diff w.r.t this x at 2
+        val y = deriv { x * x * x }
+        assertEquals(8.0, y.x)  //    y  = x * x * x = 8
+        assertEquals(12.0, x.d) // dy/dx = 3 * x * x = 12
+    }
+
+    @Test
+    fun testDiv() {
+        val x = ValueWithDeriv(5)
+        val y = ValueWithDeriv(2)
+        val z = deriv { x / y }
+        assertEquals(2.5, z.x)   //     z =  x / y   = 2.5
+        assertEquals(0.5, x.d)   // dz/dx =  1 / y   = 0.5
+        assertEquals(-1.25, y.d) // dz/dy = -x / y^2 = -1.25
+    }
+
+    @Test
+    fun testPow3() {
+        val x = ValueWithDeriv(2) // diff w.r.t this x at 2
+        val y = deriv { pow(x, 3) }
+        assertEquals(8.0, y.x)  //    y  = x ^ 3     = 8
+        assertEquals(12.0, x.d) // dy/dx = 3 * x ^ 2 = 12
+    }
+
+    @Test
+    fun testPowFull() {
+        val x = ValueWithDeriv(2)
+        val y = ValueWithDeriv(3)
+        val z = deriv { pow(x, y) }
+        assertApprox(8.0, z.x)           //     z = x ^ y = 8
+        assertApprox(12.0, x.d)          // dz/dx = y * x ^ (y - 1) = 12
+        assertApprox(8.0 * kotlin.math.ln(2.0), y.d) // dz/dy = x ^ y * ln(x)
+    }
+
+    @Test
+    fun testFromPaper() {
+        val x = ValueWithDeriv(3)
+        val y = deriv { 2 * x + x * x * x }
+        assertEquals(33.0, y.x)  //     y = 2 * x + x * x * x = 33
+        assertEquals(29.0, x.d)  // dy/dx = 2 + 3 * x * x = 29
+    }
+
+    @Test
+    fun testLongChain() {
+        val n = 10_000
+        val x = ValueWithDeriv(1)
+        val y = deriv {
+            var pow = ValueWithDeriv(1)
+            for (i in 1..n) pow *= x
+            pow
+        }
+        assertEquals(1.0, y.x)          //     y = x ^ n = 1
+        assertEquals(n.toDouble(), x.d) // dy/dx = n * x ^ (n - 1) = n - 1
+    }
+
+    @Test
+    fun testExample() {
+        val x = ValueWithDeriv(2)
+        val y = deriv { sqr(x) + 5 * x + 3*one }
+        assertEquals(17.0, y.x) // the value of result (y)
+        assertEquals(9.0, x.d)  // dy/dx
+    }
+
+    @Test
+    fun testSqrt() {
+        val x = ValueWithDeriv(16)
+        val y = deriv { sqrt(x) }
+        assertEquals(4.0, y.x)     //     y = x ^ 1/2 = 4
+        assertEquals(1.0 / 8, x.d) // dy/dx = 1/2 / x ^ 1/4 = 1/8
+    }
+
+    @Test
+    fun testSin() {
+        val x = ValueWithDeriv(PI / 6)
+        val y = deriv { sin(x) }
+        assertApprox(0.5, y.x)           //    y = sin(PI/6) = 0.5
+        assertApprox(kotlin.math.sqrt(3.0) / 2, x.d) // dy/dx = cos(PI/6) = sqrt(3)/2
+    }
+
+    @Test
+    fun testCos() {
+        val x = ValueWithDeriv(PI / 6)
+        val y = deriv { cos(x) }
+        assertApprox(kotlin.math.sqrt(3.0) / 2, y.x) //     y =  cos(PI/6) = sqrt(3)/2
+        assertApprox(-0.5, x.d)          // dy/dx = -sin(PI/6) = -0.5
+    }
+
+    private fun assertApprox(a: Double, b: Double) {
+        if ((a - b) > 1e-10) assertEquals(a, b)
+    }
+
+}