Data archive for equation discovery and recovery

Archive of data used to explore differential equation discovery approaches.

Notes

Each folder contains data and explanatory documents. The title also contains an abbreviation for the data source. Synthetic data - s_d, Real data - r_d.

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Examples

1. Wave equation with one spatial variable

$$\frac{\partial^{2} u}{\partial t^{2}} - \frac{1}{25} \frac{\partial^{2} u}{\partial x^{2}} = 0,$$ $$\\ 100\times100, x \in [0; 1], t \in [0; 1].$$ $$\\ bc = \{u(0, t) = 0, u(1, t) = 0\}, \\ \\ ic = \{u(x, 0) = 10000 \sin (\frac{1}{10}\cdot x \cdot (x-1))^2\}$$

2. Burgers' equation

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,$$ $$\\ 256\times256, x \in [-4000; 4000], t \in [0; 4].$$

3. Burgers' equation (reduced grid)

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,$$ $$\\ 100 \times100, x \in [-1000; 0], t \in [0; 1].$$

4. Lotka-Volterra equations

$$\begin{equation*} \begin{cases} \frac{\partial u}{\partial t} = \normalsize 0.55 \cdot u - 0.028 \cdot u \cdot v, \\\ \frac{\partial v}{\partial t} = \normalsize - 0.84 \cdot v + 0.026 \cdot u \cdot v. \end{cases} \end{equation*}$$ $$\\ t \in [0, 20], \\ u_0, \ v_0 = 30, 4.$$

5. Pendulum equations

$$\begin{equation} \begin{gathered} \begin{cases} \dot\sigma = z, \\\ \dot z = -\frac{1}{\sqrt{5}} z -\sin(\sigma) + 0.2 \quad \end{cases} \end{gathered} \end{equation}$$ $$\\ t \in [0, 20], \\ \sigma_0, \ z_0 = \frac{\pi}{2}, 0.5.$$

Algorithms under research:

• EPDE (Evolutionary Partial Differential Equations). Algorithm for the search of differential equations based on data (https://github.com/ITMO-NSS-team/EPDE)
• TEDEouS (Torch Exhaustive Differential Equation Solver). Algorithm for automated solution of differential equations based on neural networks (https://github.com/ITMO-NSS-team/torch_DE_solver)

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