Demo: 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)
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Center for Advanced Computing
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CVW material development is supported by NSF OAC awards 1854828, 2321040, 2323116 (UT Austin) and 2005506 (Indiana University)
CVW material development is supported by NSF OAC awards 1854828, 2321040, 2323116 (UT Austin) and 2005506 (Indiana University)