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Morten Hjorth-Jensen
2022-08-23 11:19:05 +02:00
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@@ -7,8 +7,8 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>15. Solving Differential Equations with Deep Learning &#8212; Applied Data Analysis and Machine Learning</title>
<link href="_static/css/theme.css" rel="stylesheet" />
<link href="_static/css/index.c5995385ac14fb8791e8eb36b4908be2.css" rel="stylesheet" />
<link href="_static/css/theme.css" rel="stylesheet">
<link href="_static/css/index.ff1ffe594081f20da1ef19478df9384b.css" rel="stylesheet">
<link rel="stylesheet"
@@ -31,7 +31,7 @@
<link rel="stylesheet" type="text/css" href="_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-variables.06eb56fa6e07937060861dad626602ad.css" />
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<script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
<script src="_static/jquery.js"></script>
@@ -41,22 +41,24 @@
<script src="_static/clipboard.min.js"></script>
<script src="_static/copybutton.js"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
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<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
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<link rel="index" title="Index" href="genindex.html" />
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<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
@@ -91,11 +93,11 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="intro.html">
Applied Data Analysis and Machine Learning, FYS-STK3155/4155 at the University of Oslo, Norway
Applied Data Analysis and Machine Learning
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
About the course
</span>
@@ -117,7 +119,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Review of Statistics with Resampling Techniques and Linear Algebra
</span>
@@ -134,7 +136,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
From Regression to Support Vector Machines
</span>
@@ -171,7 +173,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Decision Trees, Ensemble Methods and Boosting
</span>
@@ -188,7 +190,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Dimensionality Reduction
</span>
@@ -205,7 +207,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Deep Learning Methods
</span>
@@ -282,7 +284,7 @@
data-placement="left">.ipynb</button></a>
<!-- Download PDF via print -->
<button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
onClick="window.print()" data-toggle="tooltip" data-placement="left">.pdf</button>
onclick="printPdf(this)" data-toggle="tooltip" data-placement="left">.pdf</button>
</div>
</div>
@@ -300,7 +302,7 @@
</div>
<!-- Table of contents -->
<div class="d-none d-md-block col-md-2 bd-toc show">
<div class="d-none d-md-block col-md-2 bd-toc show noprint">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
@@ -403,7 +405,113 @@
</div>
<div id="main-content" class="row">
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
<!-- Table of contents that is only displayed when printing the page -->
<div id="jb-print-docs-body" class="onlyprint">
<h1>Solving Differential Equations with Deep Learning</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-exponential-decay">
15.1. Example: Exponential decay
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reformulating-the-problem">
15.2. Reformulating the problem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-descent">
15.3. Gradient descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-code-for-solving-the-ode">
15.4. The code for solving the ODE
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer">
15.5. The network with one input layer, specified number of hidden layers, and one output layer
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-population-growth">
15.5.1. Example: Population growth
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-forward-euler-to-solve-the-ode">
15.6. Using forward Euler to solve the ODE
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#solving-the-one-dimensional-poisson-equation">
15.7. Solving the one dimensional Poisson equation
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#comparing-with-a-numerical-scheme">
15.7.1. Comparing with a numerical scheme
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#partial-differential-equations">
15.8. Partial Differential Equations
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#type-of-problem">
15.8.1. Type of problem
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#network-requirements">
15.8.2. Network requirements
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-the-diffusion-equation">
15.9. Example: The diffusion equation
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd-the-full-program">
15.9.1. Setting up the network using Autograd; The full program
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#solving-the-wave-equation-with-neural-networks">
15.10. Solving the wave equation with Neural Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#resources-on-differential-equations-and-deep-learning">
15.11. Resources on differential equations and deep learning
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
@@ -1038,8 +1146,8 @@ Max absolute difference: 0.0437499
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 324.246
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Final cost: 0.119936
@@ -1264,8 +1372,8 @@ g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)}
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 0.221805
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Final cost: 0.000417932
@@ -1395,8 +1503,8 @@ extending the program that uses the network using Autograd:</p>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 0.221805
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Final cost: 0.000417932
@@ -1614,11 +1722,11 @@ g(x) = x(1 - x)\exp(x)
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 457.256
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 457.256
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Final cost: 0.00310113
@@ -1907,8 +2015,8 @@ f(x_{N_x - 2})
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 457.256
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Final cost: 0.00310113
@@ -2389,189 +2497,168 @@ Using TensorFlow results in a much better execution time. Try it!</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray
return array(a, dtype, copy=False, order=order)
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/core/fromnumeric.py:3208: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify &#39;dtype=object&#39; when creating the ndarray.
return asarray(a).size
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 41.05505310046362
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial cost: 41.05505310046363
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span>---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in &lt;module&gt;
&lt;ipython-input-9-fdf8a5d7c717&gt; in &lt;module&gt;
141 lmb = 0.01
142
--&gt; 143 P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
144
145 ## Store the results
/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in solve_pde_deep_neural_network(x, t, num_neurons, num_iter, lmb)
&lt;ipython-input-9-fdf8a5d7c717&gt; in solve_pde_deep_neural_network(x, t, num_neurons, num_iter, lmb)
118 # Let the update be done num_iter times
119 for i in range(num_iter):
--&gt; 120 cost_grad = cost_function_grad(P, x , t)
121
122 for l in range(N_hidden+1):
~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
18 else:
19 x = tuple(args[i] for i in argnum)
---&gt; 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
21 return nary_f
22 return nary_operator
~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in grad(fun, x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py in grad(fun, x)
23 arguments as `fun`, but returns the gradient instead. The function `fun`
24 should be scalar-valued. The gradient has the same type as the argument.&quot;&quot;&quot;
---&gt; 25 vjp, ans = _make_vjp(fun, x)
26 if not vspace(ans).size == 1:
27 raise TypeError(&quot;Grad only applies to real scalar-output functions. &quot;
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in make_vjp(fun, x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py in make_vjp(fun, x)
8 def make_vjp(fun, x):
9 start_node = VJPNode.new_root()
---&gt; 10 end_value, end_node = trace(start_node, fun, x)
11 if end_node is None:
12 def vjp(g): return vspace(x).zeros()
~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in trace(start_node, fun, x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py in trace(start_node, fun, x)
8 with trace_stack.new_trace() as t:
9 start_box = new_box(x, t, start_node)
---&gt; 10 end_box = fun(start_box)
11 if isbox(end_box) and end_box._trace == start_box._trace:
12 return end_box._value, end_box._node
~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in unary_f(x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in unary_f(x)
13 else:
14 subargs = subvals(args, zip(argnum, x))
---&gt; 15 return fun(*subargs, **kwargs)
16 if isinstance(argnum, int):
17 x = args[argnum]
/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in cost_function(P, x, t)
&lt;ipython-input-9-fdf8a5d7c717&gt; in cost_function(P, x, t)
78 g_t = g_trial(point,P)
79 g_t_jacobian = g_t_jacobian_func(point,P)
---&gt; 80 g_t_hessian = g_t_hessian_func(point,P)
81
82 g_t_dt = g_t_jacobian[1]
~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
18 else:
19 x = tuple(args[i] for i in argnum)
---&gt; 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
21 return nary_f
22 return nary_operator
~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in hessian(fun, x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py in hessian(fun, x)
76 def hessian(fun, x):
77 &quot;Returns a function that computes the exact Hessian.&quot;
---&gt; 78 return jacobian(jacobian(fun))(x)
79
80 @unary_to_nary
~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
18 else:
19 x = tuple(args[i] for i in argnum)
---&gt; 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
21 return nary_f
22 return nary_operator
~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in jacobian(fun, x)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py in jacobian(fun, x)
59 jacobian_shape = ans_vspace.shape + vspace(x).shape
60 grads = map(vjp, ans_vspace.standard_basis())
---&gt; 61 return np.reshape(np.stack(grads), jacobian_shape)
62
63 @unary_to_nary
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in stack(arrays, axis)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_wrapper.py in stack(arrays, axis)
86 # primitives defined in this file
87
---&gt; 88 arrays = [array(arr) for arr in arrays]
89 if not arrays:
90 raise ValueError(&#39;need at least one array to stack&#39;)
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in &lt;listcomp&gt;(.0)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_wrapper.py in &lt;listcomp&gt;(.0)
86 # primitives defined in this file
87
---&gt; 88 arrays = [array(arr) for arr in arrays]
89 if not arrays:
90 raise ValueError(&#39;need at least one array to stack&#39;)
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in vjp(g)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py in vjp(g)
12 def vjp(g): return vspace(x).zeros()
13 else:
---&gt; 14 def vjp(g): return backward_pass(g, end_node)
15 return vjp, end_value
16
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in backward_pass(g, end_node)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py in backward_pass(g, end_node)
19 for node in toposort(end_node):
20 outgrad = outgrads.pop(node)
---&gt; 21 ingrads = node.vjp(outgrad[0])
22 for parent, ingrad in zip(node.parents, ingrads):
23 outgrads[parent] = add_outgrads(outgrads.get(parent), ingrad)
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in &lt;lambda&gt;(g)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py in &lt;lambda&gt;(g)
65 &quot;VJP of {} wrt argnum 0 not defined&quot;.format(fun.__name__))
66 vjp = vjpfun(ans, *args, **kwargs)
---&gt; 67 return lambda g: (vjp(g),)
68 elif L == 2:
69 argnum_0, argnum_1 = argnums
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in &lt;lambda&gt;(g)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py in &lt;lambda&gt;(g)
421 A_ndim = anp.ndim(A)
422 B_meta = anp.metadata(B)
--&gt; 423 return lambda g: matmul_adjoint_1(A, g, A_ndim, B_meta)
424
425 defvjp(anp.matmul, matmul_vjp_0, matmul_vjp_1)
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in matmul_adjoint_1(A, G, A_ndim, B_meta)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py in matmul_adjoint_1(A, G, A_ndim, B_meta)
408 else: # We need to swap the last two axes of A
409 A = anp.swapaxes(A, A_ndim - 2, A_ndim - 1)
--&gt; 410 result = anp.matmul(A, G)
411 if B_is_vec:
412 result = anp.squeeze(result, anp.ndim(G) - 1)
~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
42 parents = tuple(box._node for _ , box in boxed_args)
43 argnums = tuple(argnum for argnum, _ in boxed_args)
44 ans = f_wrapped(*argvals, **kwargs)
---&gt; 45 node = node_constructor(ans, f_wrapped, argvals, kwargs, argnums, parents)
---&gt; 44 ans = f_wrapped(*argvals, **kwargs)
45 node = node_constructor(ans, f_wrapped, argvals, kwargs, argnums, parents)
46 return new_box(ans, trace, node)
47 else:
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in __init__(self, value, fun, args, kwargs, parent_argnums, parents)
34 raise NotImplementedError(&quot;VJP of {} wrt argnums {} not defined&quot;
35 .format(fun_name, parent_argnums))
---&gt; 36 self.vjp = vjpmaker(parent_argnums, value, args, kwargs)
37
38 def initialize_root(self):
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
35 @wraps(f_raw)
36 def f_wrapped(*args, **kwargs):
---&gt; 37 boxed_args, trace, node_constructor = find_top_boxed_args(args)
38 if boxed_args:
39 argvals = subvals(args, [(argnum, box._value) for argnum, box in boxed_args])
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in vjp_argnums(argnums, ans, args, kwargs)
75 &quot;VJP of {} wrt argnums 0, 1 not defined&quot;.format(fun.__name__))
76 vjp_0 = vjp_0_fun(ans, *args, **kwargs)
---&gt; 77 vjp_1 = vjp_1_fun(ans, *args, **kwargs)
78 return lambda g: (vjp_0(g), vjp_1(g))
79 else:
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in matmul_vjp_1(ans, A, B)
420 def matmul_vjp_1(ans, A, B):
421 A_ndim = anp.ndim(A)
--&gt; 422 B_meta = anp.metadata(B)
423 return lambda g: matmul_adjoint_1(A, g, A_ndim, B_meta)
424
~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
59 def f_wrapped(*args, **kwargs):
60 argvals = map(getval, args)
---&gt; 61 return f_raw(*argvals, **kwargs)
62 f_wrapped._is_primitive = True
63 return f_wrapped
~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in metadata(A)
146 @notrace_primitive
147 def metadata(A):
--&gt; 148 return _np.shape(A), _np.ndim(A), _np.result_type(A), _np.iscomplexobj(A)
149
150 @notrace_primitive
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py in find_top_boxed_args(args)
68 top_node_type = None
69 for argnum, arg in enumerate(args):
---&gt; 70 if isbox(arg):
71 trace = arg._trace
72 if trace &gt; top_trace:
KeyboardInterrupt:
</pre></div>
@@ -2900,54 +2987,42 @@ g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t)
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