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Category: A Resource – Equation

New! The YouTube Vid Series: Backpropagation and More

New! The YouTube Vid Series: Backpropagation and More

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If you are branding yourself as an AI/neural networks/deep learning person, how well do you really know the backpropagation derivation? That is, could you work through that derivation, on your own, without having to find a tutorial on it? If not – you’re in good company. MOST people haven’t worked through that derivation – and for good reason. MOST people don’t remember their chain rule methods from first semester calculus. (It probably doesn’t help if I say that the backprop…

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Filling Out the Phase Space Boundaries – 2-D CVM

Filling Out the Phase Space Boundaries – 2-D CVM

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Configuration Variables Along the Phase Space Boundaries for a 2-D CVM   Last week’s blog showed how we could get x1 for a specific value of epsilon0, by taking the derivative of the free energy and setting it equal to zero. (This works for the special case where epsilon1 is zero, meaning that there is no interaction enthalpy.) Last week, we looked at one case, where epsilon0 = 1.0. This week, we take a range of epsilon0 values and find…

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Obvious, But Useful (Getting the Epsilon-0 Value when the Interaction Enthalpy Is Zero)

Obvious, But Useful (Getting the Epsilon-0 Value when the Interaction Enthalpy Is Zero)

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  This Really Is Kind of Obvious, But …   There’s something very interesting that we can do to obtain values for the epsilon0 parameter. Let’s stay with the case where there is no interaction enthalpy. In that case, we want to find the epsilon0 value that corresponds to the x1 value at a given free energy minimum. Or conversely, given an epsilon0 value, can we identify the x1 where the free energy minimum occurs? Turns out that, for this…

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An Interesting Little Thing about the CVM Entropy (with Code)

An Interesting Little Thing about the CVM Entropy (with Code)

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The 2-D CVM Entropy and Free Energy Minima when the Interaction Enthalpy Is Zero:   Today, we transition from deriving the equations for the Cluster Variation Method (CVM) entropies (both 1-D and 2-D) to looking at how these entropies fit into the overall context of a free energy equation. Let’s start with entropy. The truly important thing about entropy is that it gives shape and order to the universe. Now, this may seem odd to those of us who’ve grown…

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Expressing Total Microstates (Omega) for the 1-D and 2-D CVM – Part 2

Expressing Total Microstates (Omega) for the 1-D and 2-D CVM – Part 2

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Showing How the Omega Equation is Obtained for the 1-D and 2-D CVM – Part 2:   The cluster variation method (CVM) lets us characterize a system in terms of local patterns, and not just the numbers of units in on (A) and off (B) states. It works with a more complex entropy term. The natural question is: How do obtain this more complex entropy? This post continues a discussion begun in the last post, on how we actually get…

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Expressing Total Microstates (Omega) for the 1-D Cluster Variation Method – Part 1

Expressing Total Microstates (Omega) for the 1-D Cluster Variation Method – Part 1

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The 1-D CVM – A Single Zigzag Chain – Part 1:   The cluster variation method (CVM) lets us characterize a system in terms of local patterns, and not just the numbers of units in on/off states. This is likely to be useful for machine learning and AI applications. Up until now, we’ve not told the story of how we actually compute the CVM entropy from the microstates. We’ll do that starting with this post; it will be a handy…

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