Fix/reorganize feature library (#74)

* Update ode_utils.py

* Update ode_solver.py

rk4 revised and __main__ added

* Delete ngclearn/utils/diffeq/ode_functions.py

* Create odes.py

odes name and structure changed

* Update __init__.py

* Create feature_library.py

* Create __init__.py

* Create base.py

* Delete docs/museum/pc-sindy.md

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* Create sindy.md

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* fix: correct feature library path and directory name

* Delete ngclearn/utils/dymbolic_dictionary directory

Author: Faezehabibi

docs/images/museum/sindy/cubic_2D.png

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+# Sparse Identification of Non-linear Dynamical Systems (SINDy)[1]
+
+In this section, we teach, create, simulate, and visualize SINDy model implemented in NGC-Learn library. After going through this demonstration, you will:
+
+1.  Learn how to build a SINDy model of time-series dataset, generated using Ordinary Differential Equations (ODE) of known dynamical systems used in [1].
+2.  Learn how to build polynomial libraries of given dataset with arbitrary order.
+3.  Learn how to solve the sparse regression problem by iteratively performing the least squares (LSQ) method followed by thresholding-- Sequential Thresholding Least Square (STLSQ)-- for the given model.
+
+The model **code** for this exhibit can be found [here](https://github.com/NACLab/ngc-museum/exhibits/sindy/sindy.py).
+
+
+## SINDy 
+SINDy is a data-driven algorithm that discovers the differential equation governing the dynamical systems. It uses symbolic regression to identify differential equation of the system and it solves sparse regression over the pre-defined library of candidate terms. It takes time series gathered dataset of the system and it gives you its describing differential equation.
+
+
+<p align="center">
+  <img src="../images/museum/sindy/sindy.png" width="700">
+</p>
+
+
+
+
+### Inputs
+> Time: $ts = [t_0, t_1, \dots,  T]$
+> State matrix: $\mathbf{X}_{(m \times n)}$  (t measurements of n variables)
+
+### Inputs
+
+   **Time**
+* $ts = [t_0,~t_1, \dots,~T]$
+  
+   **State matrix**
+* $\mathbf{X}(t)_{(m \times n)} = [x(t),~~y(t),~~z(t)]$
+
+
+
+
+
+Given a set of time-series measurements of a dynamical system state variables ($\mathbf{X}_{(m \times n)}$) we construct:
+
+
+
+
+
+
+### Data Matrix
+
+Given a set of time-series measurements of a dynamical system state variables ($\mathbf{X}_{(m \times n)}$) we construct:
+Derivative matrix: $\dot{\mathbf{X}}_{(m \times n)}$ (computed numerically)
+
+Library of Candidate Functions: $\Theta(\mathbf{X}) = [\mathbf{1} \quad \mathbf{X} \quad \mathbf{X}^2 \quad \mathbf{X}^3 \quad \sin(\mathbf{X}) \quad \cos(\mathbf{X}) \quad ...]$
+
+------------------
+
+
+<p align="center">
+  <img src="../images/museum/sindy/lorenz.png" width="300">
+  <img src="../images/museum/sindy/oscillator.png" width="300">
+</p>
+
+<p align="center">
+  <img src="../images/museum/sindy/linear_2D.png" width="300">
+  <img src="../images/museum/sindy/cubic_2D.png" width="300">
+  <img src="../images/museum/sindy/linear_3D.png" width="300">
+</p>
+
+
+
+
+
+
+SINDy describes the derivative (linear operation acting on △t) as linear transformations
+of a manually constructed dictionary from the state vector by a coefficient matrix.
+Dictionary learning combined with LASSO (L1-norm) promotes the sparsity of the coefficient matrix
+which allows only governing terms in the dictionary stay non-zero.
+
+
+
+## References
+<b>[1]</b> Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci U S A. 2016 Apr 12;113(15):3932-7. doi: 10.1073/pnas.1517384113. Epub 2016 Mar 28. PMID: 27035946; PMCID: PMC4839439.
+
+

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+from . import ode_solver
+from . import odes
+from . import ode_utils