DeepONet Derivative🔗
DeepONet Concept: Derivative as Linear Combination of Basis Functions
Enter a cubic polynomial and see how its derivative (always quadratic) can be expressed as a linear combination of simple basis functions: constant, linear, and quadratic terms.
DeepONet Concept: Basis Function Decomposition
Operator Learning: Polynomial → Derivative
Input: f(x) = ax³ + bx² + cx + d
↓ (Differentiation Operator)
Output: f'(x) = 3ax² + 2bx + c
DeepONet learns this operator!
1. Analytical Derivative:
f'(x) = 3ax² + 2bx + c
Direct differentiation of cubic polynomial → quadratic result
2. Basis Function Decomposition:
f'(x) = w₁ × 1 + w₂ × x + w₃ × x²
Any quadratic can be written as combination of: constant, linear, quadratic basis
3. DeepONet Mapping:
Branch Network: [a,b,c,d] → [w₁, w₂, w₃]
Trunk Network: x → [1, x, x²]
Branch learns coefficients, Trunk learns basis functions
4. Perfect Match:
w₁ = c, w₂ = 2b, w₃ = 3a
For this operator, the mapping is analytical and exact!
f(x) = 1.0x³ + 0.5x² - 0.3x + 0.2
f'(x) = 3.0x² + 1.0x - 0.3
f'(x) = (-0.3) × 1 + (1.0) × x + (3.0) × x²
DeepONet Basis Weights (Branch Network Output):
w₁ (Constant)
-0.3
w₂ (Linear)
1.0
w₃ (Quadratic)
3.0
Input Polynomial f(x)
Basis Functions: 1, x, x²
Weighted Basis Components
Final Result: f'(x) = Σ wᵢ × φᵢ(x)