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Applied Linear Algebraic Concepts

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Prompt for this blog entry:

Find an article or other resource about the application of linear algebra concepts in the real world. For instance, search for the real-world applications of eigenvectors and eigenvalues, matrix algebra, linear transformations, or any other linear algebra topic.

The article that I have selected to cover is, "The MLIP package: moment tensor potentials with MPI and active learning." Links to this article can be found at the end of the post in the sources section.

How is the topic being applied now?

The topic that I am choosing to discuss is centered around machine learning, and how linear algebra can be applied to machine learning in the context of material science.

I recently wrote about a few different articles in this same field of science, and it is incredibly interesting to me the more I learn about it. It concerns Machine-learning interatomic potentials, and how they are used in a variety of different ways to compute, simulate and predict possible outcomes of quantum mechanical interactions between atomic particles. MLIPs focus on the energy potentials, various energy states, and positions of atomic particles. This entire field of computational science is built on top of similar principles to those that we learned about in our linear algebra course. For instance, this area of science deals with moment / momentary potentials, which is to put simply, an energy potential that is represented by a specific atomic configuration.

The following is an example from one of the papers that I was recently reading through, and it talks about some of the calculations that are performed with these potentials using linear algebraic methods.

“In order to construct the basis functions Bα(alpha) we define the so-called level of moments:

levMμ,ν=2+4μ+ν.\displaystyle \begin{aligned} &\text{lev}M_{\mu,\nu} = 2 + 4\mu + \nu. \end{aligned}

For Example:

levM0,1=3,levM1,1=7,levM0,2=4,levM0,0=2\displaystyle \begin{aligned} \text{lev}M_{0,1} &= 3, &\quad \text{lev}M_{1,1} &= 7, &\quad \text{lev}M_{0,2} &= 4, &\quad \text{lev}M_{0,0} &= 2 \end{aligned}

The coefficients 2, 4, and 1 in (5) were empirically found to be optimal on a number of tests done in [25] and are fixed in the MLIP package. The level of multiplication, or more generally, contractions of a number of moments is defined by adding the levels,

Example:

levM1,02=12,levM0,04=8,levM2,03=30,lev(M1,1⋅M0,1)=10,lev(M1,2:M0,2)=12,lev((M0,3M0,2)⋅M0,1)=12,\displaystyle \begin{aligned} \text{lev}M_{1,0}^{2} &= 12, &\quad \text{lev}M_{0,0}^{4} &= 8, &\quad \text{lev}M_{2,0}^{3} &= 30, \\ \text{lev}(M_{1,1} \cdot M_{0,1}) &= 10, &\quad \text{lev}(M_{1,2} : M_{0,2}) &= 12, &\quad \text{lev}((M_{0,3}M_{0,2}) \cdot M_{0,1}) &= 12, \end{aligned}

where ‘·’ is the dot product of two vectors, ‘:’ is the Frobenius product of two matrices. As could be seen from these examples, non-scalar moments could yield scalars upon their contraction.” (Novikov et al., 2021).

This is only a small portion of the work discussed in the cited paper. But this is an example of work using methods such as dot-product calculations, forming a basis, or linear regression using a set of basis functions. This is also only speaking on the computation of potential and interactions. The implementations for the machine learning model that uses data from the calculated interaction would also rely on similar methods to those learned in our linear algebra course. The model implementation utilizes principles that we have covered or that are deeper into the topic of linear algebra than we have yet covered, such as markov chains.

What is the potential for future applications?

The potential for future applications is one that is growing as better and more complicated techniques are being developed and established. These techniques are what will continue to allow researchers to build models that have a much higher level of refinement when it comes to their ability to model and simulate real-world particle interactions.

How do you see using material about this topic in the future?

I am planning to go on to get a masters in the near future in either computational science or data science. So would consider the likelihood of using these materials being very high. I genuinely feel that I have enjoyed the topics covered in this course more than any other maths related topics in my academic history outside of quantitative chemistry and physics.

How will learning about this topic help your studies and career?

I feel that I have only just barely scratched the surface of the topics that we covered in our semester of linear algebra, and while I had heard of many of the topics we covered prior to this semester, I didn’t have any substantive grasp on them prior to working with them hands-on. I am excited to continue to get to work with linear algebra in the future hopefully, and to get to continue my education in the areas of materials based science and computation.

Sources: