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content/11-machine-learning/neural-net-derivation.md

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@@ -13,8 +13,8 @@ Let's start with our cost function:
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$$\mathcal{L}(A_{ij}) = \sum_{i=1}^{N_\mathrm{out}} (z_i - y_i^k)^2 = \sum_{i=1}^{N_\mathrm{out}}
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\Biggl [ g\biggl (\underbrace{\sum_{j=1}^{N_\mathrm{in}} A_{ij} x^k_j}_{\equiv \alpha_i} \biggr ) - y^k_i \Biggr ]^2$$
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where we'll refer to the product ${\boldsymbol \alpha} \equiv {\bf
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Ax}$ to help simplify notation. This means that ${\bf z} = g({\boldsymbol \alpha})$.
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where we'll refer to the product $\boldsymbol{\alpha} \equiv {\bf
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Ax}$ to help simplify notation. This means that ${\bf z} = g(\boldsymbol{\alpha})$.
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We can compute the derivative with respect to a single matrix
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element, $A_{pq}$ by applying the chain rule:

content/_config.yml

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title: "PHY 546: Python for Scientific Computing"
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author: <a href="https://zingale.github.io">Michael Zingale</a>
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#logo: logo.png
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copyright: "2022"
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copyright: "2022-2026"
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# Force re-execution of notebooks on each build.
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# See https://jupyterbook.org/content/execute.html
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use_repository_button: true
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extra_footer: |
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<p>
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&copy; 2023-2026; CC-BY-NC-SA 4.0
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&copy; 2022-2026; CC-BY-NC-SA 4.0
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<p>
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sphinx:

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