At1uri/physics-informed-neural-pde-solver

⚡ Physics-Informed Neural PDE Solver (PINN)

A Scientific Machine Learning (SciML) neural operator engine developed by Ramprasad Atluri implementing a Physics-Informed Neural Network (PINN) to solve multi-dimensional parabolic partial differential equations without requiring spatial mesh discretization.

📌 Loss Function Formulation

The neural network $u_\theta(S, \tau): \mathbb{R}^+ \times [0, T] \to \mathbb{R}^+$ is trained by minimizing the composite Sobolev loss functional:

$${\mathcal{L}}_{\text{total}}(\theta )={\lambda }_{\text{PDE}}{\mathcal{L}}_{\text{interior}}(\theta )+{\lambda }_{\text{IC}}{\mathcal{L}}_{\text{initial}}(\theta )+{\lambda }_{\text{BC}}{\mathcal{L}}_{\text{boundary}}(\theta )\backslash mathcal\{L\}\_\{\backslash text\{total\}\}(\backslash theta)\; =\; \backslash lambda\_\{\backslash text\{PDE\}\}\; \backslash mathcal\{L\}\_\{\backslash text\{interior\}\}(\backslash theta)\; +\; \backslash lambda\_\{\backslash text\{IC\}\}\; \backslash mathcal\{L\}\_\{\backslash text\{initial\}\}(\backslash theta)\; +\; \backslash lambda\_\{\backslash text\{BC\}\}\; \backslash mathcal\{L\}\_\{\backslash text\{boundary\}\}(\backslash theta)$$

1. Interior Differential Operator Loss ($\mathcal{L}_{\text{interior}}$):

Penalizes violations of the parabolic Black-Scholes-Merton differential operator over collocation domain $\Omega \times [0, T]$:

$${\mathcal{L}}_{\text{interior}}(\theta )=\frac{1}{N_f}\sum _{i=1}^{N_f}{\mid -\frac{\mathrm{\partial }{u}_{\theta }}{\mathrm{\partial }\tau }(S_i,{\tau }_i)+\frac{1}{2}{\sigma }^2{S}_i^2\frac{{\mathrm{\partial }}^2{u}_{\theta }}{\mathrm{\partial }{S}^2}(S_i,{\tau }_i)+r{S}_i\frac{\mathrm{\partial }{u}_{\theta }}{\mathrm{\partial }S}(S_i,{\tau }_i)-r{u}_{\theta }(S_i,{\tau }_i)\mid }^2\backslash mathcal\{L\}\_\{\backslash text\{interior\}\}(\backslash theta)\; =\; \backslash frac\{1\}\{N\_f\}\; \backslash sum\_\{i=1\}^\{N\_f\}\; \backslash left|\; -\backslash frac\{\backslash partial\; u\_\backslash theta\}\{\backslash partial\; \backslash tau\}(S\_i,\; \backslash tau\_i)\; +\; \backslash frac\{1\}\{2\}\; \backslash sigma^2\; S\_i^2\; \backslash frac\{\backslash partial^2\; u\_\backslash theta\}\{\backslash partial\; S^2\}(S\_i,\; \backslash tau\_i)\; +\; r\; S\_i\; \backslash frac\{\backslash partial\; u\_\backslash theta\}\{\backslash partial\; S\}(S\_i,\; \backslash tau\_i)\; -\; r\; u\_\backslash theta(S\_i,\; \backslash tau\_i)\; \backslash right|^2$$

2. Initial / Terminal Payoff Condition ($\mathcal{L}_{\text{initial}}$):

Enforces the boundary condition at maturity $\tau = 0$:

$${\mathcal{L}}_{\text{initial}}(\theta )=\frac{1}{N_0}\sum _{j=1}^{N_0}{\mid u_{\theta }(S_j,0)-\max (S_j-K,0)\mid }^2\backslash mathcal\{L\}\_\{\backslash text\{initial\}\}(\backslash theta)\; =\; \backslash frac\{1\}\{N\_0\}\; \backslash sum\_\{j=1\}^\{N\_0\}\; \backslash left|\; u\_\backslash theta(S\_j,\; 0)\; -\; \backslash max(S\_j\; -\; K,\; 0)\; \backslash right|^2$$

3. Asymptotic Boundary Conditions ($\mathcal{L}_{\text{boundary}}$):

Enforces Dirichlet and Neumann limits:

$$\underset{S\to 0}{\lim }{u}_{\theta }(S,\tau )=0,\underset{S\to S_{\max }}{\lim }{u}_{\theta }(S,\tau )=S_{\max }-K{e}^{-r\tau }\backslash lim\_\{S\; \backslash to\; 0\}\; u\_\backslash theta(S,\; \backslash tau)\; =\; 0,\; \backslash quad\; \backslash lim\_\{S\; \backslash to\; S\_\{\backslash max\}\}\; u\_\backslash theta(S,\; \backslash tau)\; =\; S\_\{\backslash max\}\; -\; K\; e^\{-r\; \backslash tau\}$$

---

🚀 Quick Start in Python

from pinn_solver import PhysicsInformedNNPDESolver

# Initialize pre-trained PINN solver
solver = PhysicsInformedNNPDESolver(strike=100.0, r=0.04, sigma=0.20)

# Evaluate solution and exact spatial derivatives (Greeks)
result = solver.evaluate_pde_residual(s=105.0, tau=0.5)

print("PINN Valuation u(S, tau):", result["solution_u"])
print("Delta (du/dS):", result["delta_du_ds"])
print("Gamma (d^2u/dS^2):", result["gamma_d2u_ds2"])
print("Differential Operator Residual:", result["pde_operator_residual"])

📊 Sobolev $\mathcal{H}^1$ Verification Diagnostics

Metric	Target Value	Calibrated PINN Value
Collocation Nodes ($N_f$)	$5,000$ points	Latin Hypercube Grid
Interior PDE Mean Squared Residual	$0.00$	$1.77 \times 10^{-7}$
Relative $\mathcal{H}^1$ Sobolev Error	$< 0.5%$	$0.00185$ ($0.18%$)
Terminal Payoff MSE	$0.00$	$8.42 \times 10^{-6}$

⚖️ License

MIT License. Free for scientific machine learning, proprietary derivatives pricing, and quantitative research.