Machine learning force field: Theory - Revision history

Ftran at 09:57, 21 April 2025

2025-04-21T09:57:14Z

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	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots		\ldots &\ldots &\dots &\dots &\dots\end{bmatrix}.
	\end{bmatrix}.
	</math>		</math>

Ftran at 09:56, 21 April 2025

2025-04-21T09:56:57Z

← Older revision		Revision as of 09:56, 21 April 2025
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	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\end{bmatrix}.		\ldots &\ldots &\dots &\dots &\dots
			\end{bmatrix}.
	</math>		</math>

Ftran at 09:56, 21 April 2025

2025-04-21T09:56:32Z

← Older revision		Revision as of 09:56, 21 April 2025
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	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\sum_i \frac{1}{N^{\alpha=2}_{\mathrm{a}}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \frac{1}{N^{\alpha=2}_{\mathrm{a}}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \frac{1}{N^{\alpha=2}_{\mathrm{a}}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \frac{1}{N^{\alpha=2}_{\mathrm{a}}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{x_{1}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{1}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{x_{1}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{1}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\

Ftran at 09:56, 21 April 2025

2025-04-21T09:56:08Z

← Older revision		Revision as of 09:56, 21 April 2025
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	\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\

Ftran at 09:55, 21 April 2025

2025-04-21T09:55:45Z

← Older revision		Revision as of 09:55, 21 April 2025
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	\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\

Ftran at 09:55, 21 April 2025

2025-04-21T09:55:22Z

← Older revision		Revision as of 09:55, 21 April 2025
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	\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{x_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{y_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\		\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=1}\right) &\sum_i \nabla_{z_{2}} K\left(\mathbf{X}^{\alpha=2}_{i},\mathbf{X}_{i_\mathrm{B}=2}\right) &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\
	\ldots &\ldots &\dots &\dots &\dots\\		\ldots &\ldots &\dots &\dots &\dots\\

Karsai at 08:47, 24 December 2024

2024-12-24T08:47:36Z

← Older revision		Revision as of 08:47, 24 December 2024
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	The evidence approximation can be done for any of the above-described regression methods, but we combine it only with the solutions from LU factorization since solutions via SVD are calculationally too expensive to be carried out multiple times.		The evidence approximation can be done for any of the above-described regression methods, but we combine it only with the solutions from LU factorization since solutions via SVD are calculationally too expensive to be carried out multiple times.

	=== ~~Prediction and error~~ estimates ===		== Error estimation ==

			=== Error estimates from Bayesian linear regression ===

	By using the relation		By using the relation
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	The mean vector <math>\mathbf{\phi}\mathbf{\bar w}</math> contains the results of the predictions on the dimensionless energy per atom, forces, and stress tensor.		The mean vector <math>\mathbf{\phi}\mathbf{\bar w}</math> contains the results of the predictions on the dimensionless energy per atom, forces, and stress tensor.
	The diagonal elements of <math>\mathbf{\sigma}</math>, which correspond to the variances of the predicted results, are used as the uncertainty in the prediction.		The diagonal elements of <math>\mathbf{\sigma}</math>, which correspond to the variances of the predicted results, are used as the uncertainty in the prediction.

			=== Spilling factor ===
			The spilling factor{{cite\|jinnouchi2:arx:2019}}{{cite\|miwa:prb:2016}} <math>s_{i}</math> is a measure of the overlap (or similarity) of a given structural environment on an atom <math> \mathbf{X}_{i}</math> with the local reference configurations <math> \mathbf{X}_{i_{\mathrm{B}}} </math> written as

			<math>
			s_{i}= 1 - \frac{ \sum\limits_{i_{\mathrm{B}}=1}^{N_{\mathrm{B}}} \sum\limits_{i'_{\mathrm{B}}=1}^{N_{\mathrm{B}}} K(\mathbf{X}_{i},\mathbf{X}_{i_{\mathrm{B}}}) K^{-1}(\mathbf{X}_{i_{\mathrm{B}}}, \mathbf{X}_{i'_{\mathrm{B}}}) K(\mathbf{X}_{i'_{\mathrm{B}}},\mathbf{X}_{i}) } { K(\mathbf{X}_{i},\mathbf{X}_{i}) }.
			</math>

			If <math>\mathbf{X}_{i}</math> is fully overlapping with any of the local reference configurations then the second term on the right hand side of the above equation becomes 1 and <math> s = 0 </math>. If <math>\mathbf{X}_{i}</math> has no similarities with any of the local reference configurations the second term on the right hand side of the above equations becomes 0 and <math> s = 1 </math>.

	== Sparsification ==		== Sparsification ==

Karsai at 07:13, 27 May 2024

2024-05-27T07:13:42Z

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	----		----

	~~<!--~~[[Category:Machine-learned force fields]][[Category:Theory]]~~-->~~		[[Category:Machine-learned force fields]][[Category:Theory]]

Karsai: /* Descriptors */

2024-04-24T13:25:02Z

Descriptors

← Older revision		Revision as of 13:25, 24 April 2024
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	<math>		<math>
	g\left(\mathbf{r}\right)=\frac{1}{~~\sqrt{2~~\sigma_{\mathrm{atom}}\pi}}\mathrm{exp}\left(-\frac{\|\mathbf{r}\|^{2}}{2\sigma_{\mathrm{atom}}^{2}}\right).		g\left(\mathbf{r}\right)=\frac{1}{\sigma_{\mathrm{atom}}\sqrt{2\pi}}\mathrm{exp}\left(-\frac{\|\mathbf{r}\|^{2}}{2\sigma_{\mathrm{atom}}^{2}}\right).
	</math>		</math>

Karsai at 13:55, 20 March 2024

2024-03-20T13:55:16Z

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	where <math>\theta</math> denotes the angle between two vectors <math>\mathbf{r}_{ij}</math> and <math>\mathbf{r}_{ik}</math> [see Fig. 2 (b)]. The important difference of the function <math>\rho_{i}^{(3)}</math> compared to the angular distribution function (also called power spectrum within the Gaussian Approximation Potential) used in reference {{cite\|bartok:prl:2010}} is that no self interaction is included, where <math>j</math> and <math>k</math> have the same distance from <math>i</math> and the angle between the two is zero. It can be shown {{cite\|jinnouchi:jcm:20}} that the self-interaction component is equivalent to the two body descriptors. This means that our angular descriptor is a pure angular descriptor, containing no two-body components and it cannot be expressed as linear combinations of the power spectrum. The advantage of this descriptor is that it enables us to separately control the effects of two- and three-body descriptors.		where <math>\theta</math> denotes the angle between two vectors <math>\mathbf{r}_{ij}</math> and <math>\mathbf{r}_{ik}</math> [see Fig. 2 (b)]. The important difference of the function <math>\rho_{i}^{(3)}</math> compared to the angular distribution function (also called power spectrum within the Gaussian Approximation Potential) used in reference {{cite\|bartok:prl:2010}} is that no self interaction is included, where <math>j</math> and <math>k</math> have the same distance from <math>i</math> and the angle between the two is zero. It can be shown {{cite\|jinnouchi:jcm:20}} that the self-interaction component is equivalent to the two body descriptors. This means that our angular descriptor is a pure angular descriptor, containing no two-body components and it cannot be expressed as linear combinations of the power spectrum. The advantage of this descriptor is that it enables us to separately control the effects of two- and three-body descriptors.

	=== Basis set expansion ===		=== Basis set expansion and descriptors ===
	The atomic probability density can be also expanded in terms of basis functions		The atomic probability density can be also expanded in terms of basis functions

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	</math>		</math>

			==== Angular filtering ====

	In many cases <math>\chi_{nl}</math> is multiplied with an angular filtering function{{cite\|boyd:book:2000}} <math>\eta</math> ({{TAG\|ML_IAFILT2}}), which can noticably reduce the necessary basis set size without losing accuracy in the calculations		In many cases <math>\chi_{nl}</math> is multiplied with an angular filtering function{{cite\|boyd:book:2000}} <math>\eta</math> ({{TAG\|ML_IAFILT2}}), which can noticably reduce the necessary basis set size without losing accuracy in the calculations
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	where <math>a_{\mathrm{FILT}}</math> ({{TAG\|ML_AFILT2}}) is a parameter controlling the extent of the filtering. A larger value for the parameter should provide more filtering.		where <math>a_{\mathrm{FILT}}</math> ({{TAG\|ML_AFILT2}}) is a parameter controlling the extent of the filtering. A larger value for the parameter should provide more filtering.

			==== Reduced descriptors ====

			A descriptor that reduces the number of descriptors with respect to the number of elements{{cite\|csanyi:npj:2022}} is written as

			<math>
			p_{n\nu l}^{iJ}=\sqrt{\frac{8\pi^{2}}{2l+1}} \sum\limits_{m=-l}^{l} c_{nlm}^{iJ} \sum\limits_{J'}c_{\nu lm}^{iJ'}.
			</math>

			For more details on the effects of this descriptor please see following site: {{TAG\|ML_DESC_TYPE}}.

	== Potential energy fitting ==		== Potential energy fitting ==
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	----		----

	[[Category:Machine-learned force fields]][[Category:Theory]]		<!--[[Category:Machine-learned force fields]][[Category:Theory]]-->