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

* Create m.md

* Add files via upload

* Delete docs/images/museum/sindy/m.md

* Add files via upload

* Create sindy.md

* Update sindy.md

* Update sindy.md

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* Update sindy.md

* Update sindy.md

* Update sindy.md

* Update sindy.md

* Update sindy.md

* Update sindy.md

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* Update sindy.md

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* Update sindy.md

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* Update sindy.md

* Update sindy.md

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* fix: correct feature library path and directory name

* Delete ngclearn/utils/dymbolic_dictionary directory

docs/museum/sindy.md

@@ -0,0 +1,82 @@
+# 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.
+
+

ngclearn/utils/diffeq/__init__.py

@@ -0,0 +1,3 @@
+from . import ode_solver
+from . import odes
+from . import ode_utils

ngclearn/utils/diffeq/ode_solver.py

@@ -167,19 +167,19 @@ def rk4(carry, dfx, dt, params, x_scale=1.):
 
     t, x = carry
 
-    f_1 = dfx(t, x, params)
-    t1, x1 = _step_forward(t, x, f_1, dt * 0.5, x_scale)
+    dfx_1 = dfx(t, x, params)
+    t2, x2 = _step_forward(t, x, dfx_1, dt * 0.5, x_scale)
 
-    f_2 = dfx(t1, x1, params)
-    t2, x2 = _step_forward(t, x, f_2, dt * 0.5, x_scale)
+    dfx_2 = dfx(t2, x2, params)
+    t3, x3 = _step_forward(t, x, dfx_2, dt * 0.5, x_scale)
 
-    f_3 = dfx(t2, x2, params)
-    t3, x3 = _step_forward(t, x, f_3, dt, x_scale)
+    dfx_3 = dfx(t3, x3, params)
+    t4, x4 = _step_forward(t, x, dfx_3, dt, x_scale)
 
-    f_4 = dfx(t3, x3, params)
+    dfx_4 = dfx(t4, x4, params)
 
-    _dx_dt = _sum_combine(f_1, f_2, f_3, f_4, w_f1=1, w_f2=2, w_f3=2, w_f4=1)
-    _t, _x = _step_forward(t, x, _dx_dt, dt / 6, x_scale)
+    _dx_dt = _sum_combine(dfx_1, dfx_2, dfx_3, dfx_4, w_f1=1, w_f2=2, w_f3=2, w_f4=1)
+    _t, _x = _step_forward(t, x, _dx_dt / 6, dt, x_scale)
 
     new_carry = (_t, _x)
     return new_carry, (new_carry, carry)
@@ -260,7 +260,7 @@ def scanner(carry, _):
 ########################################################################################
 if __name__ == '__main__':
     import matplotlib.pyplot as plt
-    from ode_functions import linear_2D
+    from feature_library import linear_2D
 
     dfx = linear_2D
     x0 = jnp.array([3, -1.5])

ngclearn/utils/diffeq/ode_utils.py

@@ -154,13 +154,17 @@ def step_rk2(t, x, dfx, dt, params, x_scale=1.):
     Returns:
         variable values iteratively integrated/advanced to next step (`t + dt`)
     """
-    f_1 = dfx(t, x, params)
+    dfx_1 = dfx(t, x, params)
 
-    t1, x1 = _step_forward(t, x, f_1, dt * 0.5, x_scale)
-    f_2 = dfx(t1, x1, params)
-    _t, _x = _step_forward(t, x, f_2, dt, x_scale)
+    t1, x1 = _step_forward(t, x, dfx_1, dt * 0.5, x_scale)
+    dfx_2 = dfx(t1, x1, params)
+    _t, _x = _step_forward(t, x, dfx_2, dt, x_scale)
     return _t, _x
 
+
+
+
+
 @partial(jit, static_argnums=(2, ))
 def step_rk4(t, x, dfx, dt, params, x_scale=1.):
     """
@@ -191,19 +195,20 @@ def step_rk4(t, x, dfx, dt, params, x_scale=1.):
     Returns:
         variable values iteratively integrated/advanced to next step (`t + dt`)
     """
-    f_1 = dfx(t, x, params)
-    t1, x1 = _step_forward(t, x, f_1, dt * 0.5, x_scale)
 
-    f_2 = dfx(t1, x1, params)
-    t2, x2 = _step_forward(t, x, f_2, dt * 0.5, x_scale)
+    dfx_1 = dfx(t, x, params)
+    t2, x2 = _step_forward(t, x, dfx_1, dt * 0.5, x_scale)
+
+    dfx_2 = dfx(t2, x2, params)
+    t3, x3 = _step_forward(t, x, dfx_2, dt * 0.5, x_scale)
 
-    f_3 = dfx(t2, x2, params)
-    t3, x3 = _step_forward(t, x, f_3, dt, x_scale)
+    dfx_3 = dfx(t3, x3, params)
+    t4, x4 = _step_forward(t, x, dfx_3, dt, x_scale)
 
-    f_4 = dfx(t3, x3, params)
+    dfx_4 = dfx(t4, x4, params)
 
-    _dx_dt = _sum_combine(f_1, f_2, f_3, f_4, w_f1=1, w_f2=2, w_f3=2, w_f4=1)
-    _t, _x = _step_forward(t, x, _dx_dt, dt / 6, x_scale)
+    _dx_dt = _sum_combine(dfx_1, dfx_2, dfx_3, dfx_4, w_f1=1, w_f2=2, w_f3=2, w_f4=1)
+    _t, _x = _step_forward(t, x, _dx_dt / 6, dt, x_scale)
     return _t, _x
 
 @partial(jit, static_argnums=(2, ))