---
license: mit
task_categories:
- tabular-regression
- time-series-forecasting
tags:
- physics-informed-neural-networks
- pinn
- partial-differential-equations
- quantitative-finance
- scientific-machine-learning
- sobolev-space
pretty_name: High-Dimensional PDE Collocation & Sobolev Benchmarks
size_categories:
- 1K<n<10K
---

# 📊 High-Dimensional PDE Collocation & Sobolev Benchmarks

An institutional scientific machine learning (SciML) benchmark dataset curated by **[Ramprasad Atluri](https://huggingface.co/At1uri)** for training and benchmarking **Physics-Informed Neural Networks (PINNs)** on parabolic and elliptic partial differential equations.

## 📌 Mathematical Operator Formulation

Physics-Informed Neural Networks approximate the solution $u_\theta(S, t)$ by directly penalizing the residual of the differential operator $\mathcal{L}[u]$ in Sobolev space $\mathcal{H}^2(\Omega)$:

$$\mathcal{L}[u] = -\frac{\partial u}{\partial \tau} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 u}{\partial S^2} + r S \frac{\partial u}{\partial S} - r u = 0$$

Subject to the terminal payoff condition on the boundary $\partial \Omega$:

$$u(S, 0) = \max(S - K, 0)$$

And Dirichlet / Neumann asymptotic boundary conditions:

$$\lim_{S \to 0} u(S, \tau) = 0, \quad \lim_{S \to \infty} u(S, \tau) = S - K e^{-r \tau}$$

## 🔬 Collocation Mesh Specifications

This dataset contains **5,000 Latin-hypercube spatio-temporal collocation nodes** $(S_i, \tau_i) \in [20.0, 250.0] \times [0.01, 1.5]$ equipped with analytical ground truths:

1. **Exact Functional Solution**:
   $$u(S, \tau) = S \Phi(d_1) - K e^{-r \tau} \Phi(d_2)$$
2. **First Spatial Derivative ($\Delta = \nabla_S u$)**:
   $$\Delta = \frac{\partial u}{\partial S} = \Phi(d_1)$$
3. **Second Spatial Derivative ($\Gamma = \nabla_S^2 u$)**:
   $$\Gamma = \frac{\partial^2 u}{\partial S^2} = \frac{\phi(d_1)}{S \sigma \sqrt{\tau}}$$
4. **Temporal Derivative ($\Theta = \partial u / \partial \tau$)**:
   $$\Theta = \frac{\partial u}{\partial \tau} = \frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} + r K e^{-r \tau} \Phi(d_2)$$
5. **Differential Residual ($\mathcal{R}_{\text{PDE}}$)**:
   $$\mathcal{R}(S_i, \tau_i) = -\Theta_i + \frac{1}{2} \sigma^2 S_i^2 \Gamma_i + r S_i \Delta_i - r u_i \equiv 0$$

## 🚀 Loading in Python

```python
import pandas as pd

url = "https://huggingface.co/datasets/At1uri/high-dimensional-pde-collocation-benchmarks/raw/main/pde_collocation_mesh_benchmarks.csv"
df = pd.read_csv(url)

print(df.head())
print("Maximum Analytical Residual Error:", df["pde_operator_residual"].abs().max())
```

## 🛠️ Schema Definition

| Feature | Type | Description |
| :--- | :--- | :--- |
| `point_id` | Integer | Unique collocation node index |
| `spot_s` | Float | Spatial coordinate $S \in \mathbb{R}^+$ |
| `time_to_maturity_tau` | Float | Temporal coordinate $\tau = T - t$ |
| `exact_solution_u` | Float | Analytical valuation $u(S, \tau)$ |
| `first_spatial_derivative_delta` | Float | First derivative $\Delta = \partial u / \partial S$ |
| `second_spatial_derivative_gamma`| Float | Second derivative $\Gamma = \partial^2 u / \partial S^2$ |
| `temporal_derivative_theta` | Float | Time decay $\Theta = \partial u / \partial \tau$ |
| `volatility_sensitivity_vega` | Float | Volatility sensitivity $\nu = \partial u / \partial \sigma$ |
| `pde_operator_residual` | Float | Verification residual $\mathcal{L}[u](S_i, \tau_i)$ |

## ⚖️ License
MIT License. Free for scientific machine learning, quantitative finance, and deep neural PDE research.