DOI: 10.5281/zenodo.15140827
title: Bi-level Identification of Governing Equations for Nonlinear Physical Systems

description: This data set contains all data used in the paper "Bi-level identification of complex dynamical systems through reinforcement learning".

In the BILLIE (Bi-level Identification of Equations) algorithm proposed in the paper, two sets of orthogonal data (denote as s1 and s2) were
used in each identification case.
The "_s1" (or "_s2") in a dataset's name means that s1 (or s2) was sampled from that dataset.
Those datasets without "_s1" (or "_s2") in the name means that both s1 and s2 was sampled from that dataset.
Each of the four folders is detailed as below.
1. The folder "Navier-Stokes equation" contains the simulated data of Navier-stokes equation for the three fluid dynamics identification cases.
    Naming: "NS_(2D or 3D)_(Reynolds number)_(s1/s2 if any)" 

    Case1: 2D flow with Reynolds number of 100
               The two sets of data is structured on a 256x256 grid within the 2pi x 2pi spatial domain, meaning dx=dy=2pi/256.
               The time step is dt=0.0015.
               The data is organized as [T, X, Y, C] where T is the temporal dimension, X, Y are the spatial dimensions, C=[V, U, P] where U, V are the two 
               fluid velocity components on the two spatial dimensions respectively, and P is the pressure scalar.
        
    Case2: 2D flow with Reynolds number of 1000
               The two sets of data is structured on a 2048x2048 grid within the 2pi x 2pi spatial domain, meaning dx=dy=2pi/2048.
               The time step is dt=0.00005.
               The data is organized as [T, X, Y, C], identically to case1.

    Case3: 3D flow with Reynolds number of 100
               This is the data of a flow around a cylinder published by Raissi in "Hidden fluid mechanics: Learning velocity and pressure fields from
               flow visualizations" (DOI: 10.1126/science.aaw4741)

2. The folder "Burgers' equation" contains the simulated data of Burgers' equation for the experiments on small-coefficient terms, noise, and sparsity.
    Ground truth equation: u_t = lambda*u_xx - u*u_x
    Naming: "Burgers_coef_(lambda)_(grid setting: spatial x temporal)_(s1/s2)" 
    The datasets were simulated on a [-8, 8] spatial domain and [0, 10] temporal domain with structured grids of different levels of sparsity.
    The noise tests were performed on "Burgers_coef_1e-1_257x101_s1.mat" and "Burgers_coef_1e-1_257x101_s2.mat" by manually adding gaussian noise.

3. The folder "Three body" contains the simulated data of the three-body system for the experiments on small-coefficient terms, noise, and sparsity.
    Naming: "three_body_coef_(lambda, see paper for meaning)_(s1/s2)"
    Each dataset contains a dictionary of 7 keys: x, y, z, u, v, w, dt.
    The time step is dt=0.005.
    The noise tests were performed on "three_body_coef_1e0_s1.mat" and "three_body_coef_1e0_s2.mat" by manually adding gaussian noise.

4. The folder "Single-cell sequencing data" contains the two sets of preprocessed multi-omics single-cell sequencing datasets used in identifying RNA 
    and protein velocity.
    The original datasets of GSM2695381 and GSM2695382 is publicly available in the Gene Expression Omnibus ("Large-scale simultaneous
    measurement of epitopes and transcriptomes in single cells").