Conclusion

Manuscript/stochastic_triples.bib

@@ -273,3 +273,17 @@ i@article{watson_2016,
 	publisher = {AIP Publishing},
 	doi = {10.1063/1.4992127}
 }
+
+@article{ceperley_2024,
+	author = {Ceperley, David M. and Jensen, Scott and Yang, Yubo and Niu, Hongwei and Pierleoni, Carlo and Holzmann, Markus},
+	title = {{Training models using forces computed by stochastic electronic structure methods}},
+	journal = {Electron. Struct.},
+	volume = {6},
+	number = {1},
+	pages = {015011},
+	year = {2024},
+	month = mar,
+	issn = {2516-1075},
+	publisher = {IOP Publishing},
+	doi = {10.1088/2516-1075/ad2eb0}
+}

Manuscript/stochastic_triples.tex

@@ -150,10 +150,6 @@ In the following sections of this paper, we will provide a brief introduction to
 \section{Theoretical Background}
 \label{sec:theory}
 
-%a. Brief overview of coupled cluster theory
-%b. Perturbative triples in coupled cluster theory
-%c. Challenges and limitations of traditional approaches
-
 
 The perturbative triples correction,
 \begin{equation}
@@ -190,17 +186,6 @@ In the algorithm proposed by Rendell\cite{rendell_1991}, for each given triplet
 \section{Semi-Stochastic Algorithm}
 \label{sec:algorithm}
 
-%a. Explanation of the algorithm's main principles and methodology
-% - Rewriting with nV in outer-most loops
-% - Implies summing all permutations of (a,b,c) to avoid extra dgemms
-%  - Comparison in number of Gflops to theory
-%b. Description of the stochastic sampling procedure
-%    - Memoization
-%    - Importance function
-%    - Semi-stochastic algorithm
-%c. Discussion of the algorithm's advantages and potential trade-offs
-%d. Detailed pseudocode or algorithmic steps, if applicable
-
 \subsection{Stochastic formulation}
 
 We propose an algorithm influenced by the semi-stochastic approach originally developed for computing the Epstein-Nesbet second-order perturbation correction to the energy. \cite{garniron_2017}
@@ -347,24 +332,6 @@ $t_0 \leftarrow \text{WallClockTime}()$ \;
 \end{algorithm}
 
 
-%a. Description of the computational framework and software used
-%b. Discussion of any specific optimizations or parallelization techniques employed
-% - Explain that form_w and form_v can be entirely produced by dgemm
-% - Show implementation with OpenMP tasks
-%c. Any relevant technical considerations or limitations
-% - Limitations: memory because in-core algorithm.
-
-%=================================================================%
-\section{Numerical experiments}
-
-% + Benzene TZ/QZ
-% - Streptocyanine QZ: Small molecule in a large basis set
-% - Caffeine def2-svp: Large molecule in a small basis set
-% + Vibrational frequency of CuCl/cc-pvqz
-% + Measure flops and compare to the peak
-%c. Analysis of the algorithm's accuracy, efficiency, and scalability
-%d. Discussion of any observed limitations or challenges
-
 \subsection{Convergence of the statistical error in benzene}
 
 In this section we illustrate the convergence of the statistical error of the perturbative triples correction as a function of the computational cost.
@@ -398,7 +365,20 @@ This trend underscores the algorithm's enhanced suitability for systems with few
 
 \subsection{Vibrational frequency of copper chloride}
 
-Our methodology proves especially advantageous for scenarios requiring the aggregation of numerous CCSD(T) energies, such as neural network training or the exploration of potential energy surfaces.
+Our methodology proves especially advantageous for scenarios requiring the
+aggregation of numerous CCSD(T) energies, such as neural network training or
+the exploration of potential energy surfaces.
+In a recent article, the authors highlight the pivotal role of Quantum Monte
+Carlo (QMC) in generating data for constructing potential energy surfaces.
+The study suggests that stochastic noise inherent in QMC can facilitate machine
+learning model training, demonstrating that models can benefit from numerous,
+less precise data points. These findings are supported by an analysis of
+machine learning models, where noise not only helped improve model accuracy but
+also enabled error estimation in model predictions.
+Similarly to QMC, our semi-stochastic formulation could take advantage of many
+low-accuracy points.
+
+
 In this section, we discuss the application of our novel algorithm within the context of computing vibrational frequencies, specifically through the example of copper chloride (\ce{CuCl}).
 A demonstrative application presented here involves the determination of the equilibrium bond length and the computation of the vibrational frequency of \ce{CuCl} using the CCSD(T)/cc-pVQZ level of theory.
 The procedure involves determining the CCSD(T) potential energy curve for \ce{CuCl}, followed by its analytical representation through a Morse potential fitting:
@@ -496,14 +476,22 @@ Beyond these thresholds, particularly after 64 cores on the ARM server, the heav
 \section{Conclusion}
 \label{sec:conclusion}
 
-%a. Summary of the algorithm and its advantages
-%b. Recapitulation of the key findings and contributions
-%c. Final remarks and encouragement for further research
-%
-% Works better for few electrons in large basis sets
-% Interesting for ML or PES exploration
+In this work, we introduced a semi-stochastic algorithm for accelerating the computation of the perturbative triples correction in coupled cluster calculations.
+This novel approach combines deterministic and stochastic methods to optimize both accuracy and computational efficiency.
+The core of our algorithm is based on selectively calculating contributions labeled by triplets of virtual orbitals leveraging Monte Carlo sampling, and employing memoization to suppress redundant calculations.
+
+Our results demonstrate that the semi-stochastic algorithm substantially reduces the computational effort compared to traditional deterministic methods, achieving near-exact accuracy with significantly reduced computational resources. Specifically, we have shown that the algorithm can achieve chemical accuracy with a small fraction of the computational effort required by fully deterministic approaches. This efficiency opens up new possibilities for studying larger systems or employing more extensive basis sets that were previously beyond reach due to computational constraints.
+Additionally, the implementation of this algorithm has proven to be highly parallelizable, demonstrating excellent scalability across different high-performance computing platforms.
+
+An important aspect of our investigation focused on the application of our algorithm to potential energy surface scanning.
+Our method aligns well with recent findings suggesting the utility of numerous, less precise data points in constructing machine learning models.\cite{ceperley_2024}
+For instance, we demonstrated that fitting a PES using data points generated with relatively large error bars using our algorithm still resulted in highly accurate values for the vibrational frequency and the equilibrium distance of copper chloride.
+This capability to produce large datasets with controlled accuracy efficiently will be particularly advantageous for training machine learning models that are robust to variations in input data quality.
+
+
+Therefore, our semi-stochastic algorithm not only addresses the challenge of computational expense in quantum chemistry calculations but also facilitates the generation of extensive datasets needed for machine learning applications.
+This method holds significant potential for advancing computational studies in chemistry, particularly in dynamic simulations and large-scale electronic structure investigations. We advocate for continued exploration of this methodology to expand its application to other computationally demanding tasks in quantum chemistry and to explore further integration into machine learning-based chemical research.
 
-%=================================================================%