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Magnetic multi-component moment tensor potential (mMTP)

The concept of magnetic multi-component Moment Tensor Potential (mMTP) presented in the current research is based on the previously developed non-magnetic MTP for multi-component systems41,42 and magnetic MTP for single-component systems35.

The mMTP potential is local, i.e., the energy of the atomistic system is a sum of energies of individual atoms:

$$begin{aligned} E = sum _{i=1}^{N_a}E_i, end{aligned}$$

(1)

where i stands for the individual atoms in an (N_a)-atom system. We note that any configuration includes lattice vectors ({{varvec{L}}} = {{{varvec{l}}}_1,{{varvec{l}}}_2,{{varvec{l}}}_3}), atomic positions ({{varvec{R}}} = {{{varvec{r}}}_1, ldots , {{varvec{r}}}_{N_a}}), types (Z = {z_1,ldots ,z_{N_{a}}}) (we also denote (N_{rm types}) by the total number of atomic types in the system), and magnetic moments (M = {m_1,ldots ,m_{N_a}}). The energy of the atom (E_i), in turn, has the form:

$$begin{aligned} E_i = sum _{alpha =1}^{alpha _{rm max}} xi _{alpha }B_{alpha }({mathfrak n}_i), end{aligned}$$

(2)

where ({{varvec{xi }}} = {xi _{alpha } }) are the linear parameters to be optimized and (B_alpha) are the so-called basis functions, which are contractions of the descriptors25 of atomistic environment ({mathfrak n}_i), yielding a scalar. The (alpha _text {max}) parameter can be changed to provide potentials with different amount of parameters35.

The descriptors are composed of the radial part, i.e., the scalar function depending on the interatomic distances and atomic magnetic moments, and the angular part, which is a tensor of rank (nu):

$$begin{aligned} M_{mu ,nu }({mathfrak n}_i)=sum _{j} f_{mu }(| {{varvec{r}}}_{ij}|,z_i,z_j,m_i,m_j)underbrace{{{varvec{r}}}_{ij}otimes ...otimes {{varvec{r}}}_{ij}}_nu text { times }, end{aligned}$$

(3)

where ({mathfrak n}_i) stands for the atomic environment, including all the atoms within the (R_text {cut}) distance (or less) from the central atom i, (mu) is the number of the radial function, (nu) is the rank of the angular part tensor, (|{{varvec{r}}}_{ij}|) is the distance between the atoms i and j, (z_i) and (z_j) are the atomic types, (m_i) and (m_j) are the magnetic moments of the atoms.

The radial functions are expanded in a basis of Chebyshev polynomials:

$$begin{aligned} f_{mu }(|r_{ij}|,z_i,z_j,m_i,m_j) = sum _{zeta =1}^{N_{phi }} sum _{beta =1}^{N_{psi }}sum _{gamma =1}^{N_{psi }}c_{mu ,z_i,z_j}^{zeta ,beta ,gamma } phi _{zeta }(|{varvec{r}}_{ij}|) psi _{beta }(m_i)psi _{gamma }(m_j) (R_{rm cut} - |{varvec{r}}_{ij}|)^2. end{aligned}$$

(4)

Here ({{varvec{c}}} = {c_{mu ,z_i,z_j}^{zeta ,beta ,gamma }}) are the radial parameters to be optimized, each of the functions (phi _{zeta }(|{varvec{r}}_{ij}|)), (psi _{beta }(m_i)), (psi _{gamma }(m_i)) is a Chebyshev polynomial of order (zeta), (beta) and (gamma) correspondingly, taking values from (-1) to 1. The function (phi _{zeta }(|{varvec{r}}_{ij}|)) yields the dependency on the distance between the atoms i and j, while the functions (psi _{beta }(m_i)) and (psi _{gamma }(m_j)) yield the dependency on the magnetic moments of the atoms i and j, correspondingly. The arguments of the functions (phi _{zeta }(|{varvec{r}}_{ij}|)) are on the interval ((R_{rm min},R_{rm cut})), where (R_{rm min}) and (R_{rm cut}) are the minimum and maximum distance, correspondingly, between the interacting atoms. The functions (psi _{beta }(m_i)) and (psi _{gamma }(m_j)) are of the same structure, which we explain for the case of the former one. The argument of the function (psi _{beta }(m_i)) is the magnetic moment of the atom i, taking the values on the ((-M_{rm max}^{z_i},M_{rm max}^{z_i})) interval. The value (M_{rm max}^{z_i}) itself depends on the type of atom (z_i), and is determined as the maximal absolute value of the magnetic moment for atom type (z_i) in the training set. Similar to the conventional MTP, the term ((R_{rm cut} - |{varvec{r}}_{ij}|)^2) provides smooth fading to 0 when approaching the (R_{rm cut}) distance, in accordance with the locality principle (1).

We note that magnetic degrees of freedom (m_i) from (4) are collinear, i.e., they can take negative or positive values as projection onto the Z axis (though the choice of the axis is arbitrary). This way, in comparison to non-magnetic atomistic systems with N atoms, in which the amount of degrees of freedom equals 4N (namely 3N for coordinates and N for types), for the description of magnetic systems additional N degrees of freedom are introduced, standing for the magnetic moment (m_i) of each atom. The amount of parameters entering the radial functions (Eq. 4) also increases in mMTP compared to the conventional MTP41,42. Namely, in MTP this number equals (N_{mu } cdot N_{phi } cdot N_{rm types}^2), while in mMTP it is (N_{mu } cdot N_{phi } cdot N_{rm types}^2 cdot N_{psi }^2). Thus, if we take (N_{psi } = 2) (which is used in the current research), the amount of the parameters entering the radial functions would be four times more in mMTP then in MTP.

We denote all the mMTP parameters by ({varvec{theta }}= {{varvec{xi }}, {varvec{c}} }) and the total energy (1) of the atomic system by (E=E({{varvec{theta }}})=E({{varvec{theta }}};M)=E({{varvec{theta }}};{{varvec{L}}},{{varvec{R}}},Z,M)).

The tensor (Eq. (4)) includes collinear magnetic moments in its functional form. However, it is not invariant with respect to the inversion of magnetic moments, i.e., (E({{varvec{theta }}};M) ne E({{varvec{theta }}};-M)), while both original and spin-inverted configurations should yield the same energy due to the arbitrary orientation of the projection axis, which we further call the magnetic symmetry.

We use data augmentation followed by explicit symmetrization with respect to magnetic moments to train a symmetric mMTP as we discuss below. Assume we have K configurations in the training set with DFT energies (E_k^{rm DFT}), forces ({varvec{f}}^{rm DFT}_{i,k}), and stresses (sigma ^{rm DFT}_{ab,k}) ((a,b=1,2,3)) calculated. We find the optimal parameters (bar{{{varvec{theta }}}}) (fit mMTP) by minimizing the objective function:

$$begin{aligned} &sum _{k=1}^{K} Biggl [ w_{rm e} Biggl | frac{E_k ({varvec{theta }}; M) + E_{k}({varvec{theta }}; -M)}{2} - E_{k}^{rm DFT}Biggr |^2 \&quad + w_{rm f} sum _{i=1}^{N_a} Biggl | frac{{varvec{f}}_{i,k}({varvec{theta }};M) + {varvec{f}}_{i,k}({varvec{theta }};-M)}{2} - {varvec{f}}^{rm DFT}_{i,k}Biggr |^2 \&quad +w_{rm s} sum _{a,b=1}^{3} Biggl | frac{sigma _{ab,k}({varvec{theta }};M)+sigma _{ab,k}({varvec{theta }};-M)}{2} -sigma ^{rm DFT}_{ab,k}Biggr |^2 Biggr ], end{aligned}$$

(5)

where (w_{rm e}), (w_{rm f}), and (w_{rm s}) are non-negative weights. By minimizing (5) we find such optimal parameters (bar{{{varvec{theta }}}}) that yield (E_k (bar{{varvec{theta }}}; M) approx E_k (bar{{varvec{theta }}}; -M)), (k = 1, ldots , K) (the same fact takes place for the mMTP forces and stresses), i.e., we symmetrize the training set to make mMTP learn the required symmetry from the data itselfthis is called data augmentation.

Next, we modify mMTP to make the energy used for the simulations (e.g., relaxation of configurations) to satisfy the exact symmetry:

$$begin{aligned} E^{rm symm}(bar{{{varvec{theta }}}};M) = dfrac{E(bar{{varvec{theta }}};M)+E(bar{{varvec{theta }}};-M)}{2}. end{aligned}$$

(6)

That is, we substitute the mMTP energy (1) into (6) and get a functional form which satisfies the exact identity (E^{rm symm}(bar{{{varvec{theta }}}};M) = E^{rm symm}(bar{{{varvec{theta }}}};-M)) for any configuration. We also note that (E (bar{{varvec{theta }}}) approx E^{rm symm}(bar{{{varvec{theta }}}})).

We use the cDFT approach with hard constraints(i.e., Lagrange multiplier) as proposed by Gonze et al. in Ref.19. One way to formulate it is to first note that in a single-point DFT calculation we minimize the Kohn-Sham total energy functional (E[rho ; {{varvec{R}}}]) with respect to the electronic density (rho =rho (r)) (here (rho) combines the spin-up and spin-down electron densities), keeping the nuclei position ({{varvec{R}}}) fixed. In other words, we solve the following minimization problem:

$$begin{aligned} E_{rm DFT}({{varvec{R}}}) = min _rho E[rho ; {{varvec{R}}}], end{aligned}$$

and from the optimal (rho ^* = mathrm{arg,min} E[rho ; {{varvec{R}}}]) we can, e.g., find magnetization (m(r) = rho ^*_+ - rho ^*_-), where the subscripts denote the spin-up ((+)) and spin-down () densities. The magnetic moment of the ith atom can be found by integrating m(r) over some (depending on the partitioning scheme) region around the atom:

$$begin{aligned} m_i = int _{Omega _i} m(r) textrm{d}r. end{aligned}$$

(7)

Since the minimizer (rho ^*) depends on ({{varvec{R}}}), (m_i) are also the functions of ({{varvec{R}}}).

According to the cDFT approach19, we now formulate the problem of minimizing (E[rho ; {{varvec{R}}}]) in which not only ({{varvec{R}}}),but also (rho) is allowed to change only subject to constraints (7):

$$begin{aligned} begin{array}{rcl} E_{rm cDFT}(rho, {{varvec{R}}}, M) =&{} min _rho &{} E[rho ; {{varvec{R}}}] \ &{} text {subject to} &{} m_i = int _{Omega _i} big (rho _{+}(r)-rho _-(r)big ) textrm{d}r. end{array} end{aligned}$$

The algorithmic details of how this minimization problem is solved, and how the energy derivatives (forces, stresses, torques) are computed, are described in detail in Ref.19.

We used the ABINIT code43,44 for DFT (and cDFT recently developed and described in Ref.19) calculations with (6times 6times 6) k-point mesh and cutoff energy of 25 Hartree (about 680 eV). We utilized the PAW PBE method with the generalized gradient approximation. We applied constraints on magnetic moments of all atoms during cDFT calculations.

We fitted an ensemble of five mMTPs with 415 parameters in order to quantify the uncertainty of mMTPs predictions. For each mMTP we took (R_{rm min} = 2.1 ~) , (R_{rm cut} = 4.5 ~), (M_{rm max}^{rm Al} = 0.1 ~mu _B), and (M_{rm max}^{rm Fe} = 3.0 ~mu _B). The weights in the objective function (5) were (w_{rm e} = 1), (w_{rm f} = 0.01) (^2), and (w_{rm s} = 0.001).

See the article here:
Constrained DFT-based magnetic machine-learning potentials for magnetic alloys: a case study of FeAl | Scientific ... - Nature.com

A research team have built a virtual creature mimicking the many brain-containing limbs of an octopus.

Hard, clunky robots often struggle to get around the world as well as animals do. Soft, bendable materials, which better simulate natural muscles, are hailed by roboticists as the key to building more adaptable machines, but because they can move in so many different ways, they are extremely challenging to control.

Evolution already figured out how to control soft materials, so Mattia Gazzola, a mechanical engineering professor at the University of Illinois Urbana-Champaign, turned to nature for inspiration. All sorts of creatures play tricks to minimize their computing requirements, he said. Theres this mechanical intelligence in the body itself.

No animal embodies the meticulous coordination of soft limbs like the octopus. Theyre known for their intelligence and creativity, driven not by one brain, but many.

Octopuses have a highly distributed nervous system, with one brain housed in their mantle (the animals blob-like body) performing high-level functions like learning and decision-making, and neural tissue in each limb running more basic motor commands on their own.

Inspired by this hierarchical brain structure a central controller managing actions powered by the limbs Gazzolas team, led by Ph.D. student Chia-Hsien Shih, built an octopus of their own. In a study published in Advanced Intelligent Systems, they present the CyberOctopus, a simulated multi-limbed soft robot that harnesses a hierarchical machine learning strategy to forage for virtual treats.

Each arm of the CyberOctopus was modeled as a bendable rod enveloped by elastic virtual muscles. By activating different combinations of these muscles, the limb can contract, bend, extend, or twist. Traveling waves of muscle contractions can undulate the arms to pull the creature across the floor of a virtual environment, or grab a food target and bring it to its mouth.

One common approach in machine learning and robotics, Gazzola said, is to throw a huge neural network at the system and hope it learns what to do. This can work in simple environments where there are a limited number of possible actions and outcomes for the robot to experience. But here, he said, there are too many variables to deal with. We tried, and it just doesnt work.

Instead, Shih and Gazzola designed a three-tiered control system to guide the CyberOctopus. The lowest level deals with obstacles with basically no computation at all, instructing limb muscles to reflexively relax when they bump into something.

Above that, each individual limb employs two simple algorithms that allow it to do two basic behaviors on its own: reach for a food target and crawl. At the top, a more complex algorithm tries to create an optimal sequence of those two behaviors reach and crawl to gather as much food as possible while using as little energy as possible.

After confirming that they could activate the muscles required to make the model crawl around a virtual environment, they challenged it with progressively harder food-gathering tasks.

The researchers tracked how much energy the octopus spent relative to how much it regained by eating, and they found that their hierarchical control technique successfully guided the CyberOctopus through its foraging challenges all without relying on massive neural networks.

Theres a tendency nowadays to use neural networks for everything, Gazzola said. They are very powerful tools. But if you understand the physical problem, then you can leverage that understanding to your advantage.

While the CyberOctopus succeeded at the tasks it faced here, the model still doesnt hold a candle to the creative problem-solving abilities of a real octopus. Robots capable of figuring out, on the fly, how to escape from a virtual aquarium in such a big parameter space, in a complex environment, are not there yet, Gazzola said.

But materials science, robotics, and machine learning techniques are getting more powerful by the day, so a CyberOctopus imbued with true octopus-like intelligence may be possible down the line.

Moving forward, Gazzola dreams of building bio-hybrid soft machines, which will perform computations using both synthetic and living tissues. The octopus was an excuse to develop this technology [for broader use], he said.

For example, soft robots may be especially well-suited for navigating harsh environments where standard rigid robots struggle, like these tube-like inflatable robots that can squeeze through tight spots and lift heavy objects. And softness is always the goal for robots that will interact with humans in potential medical, caretaking, or emergency response roles.

Its a vast space, Gazzola said, with a wide range of future applications to explore. If you understand the body, you can use it to solve an otherwise difficult problem in a simple fashion. This demonstrates that this is possible.

Reference: C Shih, et al., Hierarchical Control and Learning of a Foraging CyberOctopus, Advanced Intelligent Systems (2023). DOI: 10.1002/aisy.202300088

Continued here:
Meet CyberOctopus, your guide to the world of machine learning ... - Advanced Science News

To combat fraudulent sales of low-aged ginseng disguised as high-aged ginseng, scientists from Shanghai University of Traditional Chinese Medicine in China, developed machine learning (ML) models to predict the ages of ginseng samples. Their work was published in the Journal of Separation Science (1).

The study aims to differentiate mountain-cultivated ginseng by age. Ginseng has been studied for its multiple health benefits, including boosting the immune system and fighting conditions like colds, flu, and cancer. Mountain-cultivated ginseng is usually harvested after 10 years and can produce more berries and seeds than wild ginseng plants, or ginseng plants harvested after 15 years. This difference has led to issues in the ginseng market, with some fraudulently selling low-aged cultivated ginseng and disguising it as high-aged.

For this experiment, 98 ginseng samples were analyzed using liquid chromatographymass spectrometry (LCMS), with multivariate statistical analysis used to identify patterns between samples and influential components. ML models were also created to assist in this process. Untargeted metabolomic analysis divided samples that were between 420 years into three age groups, with 22 age-dependent biomarkers discovered to differentiate between said age groups.

From there, three more ML models were made to predict new samples, which eventually led to an optimal model being selected. According to the scientists, Some biomarkers could determine age phases according to the differentiation of mountain-cultivated ginseng samples (1). The biomarkers were later analyzed for potential variation trends.

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Machine Learning Models Help Researchers Predict the Ages of ... - LCGC Chromatography Online

Last fall, the artist and musician Holly Herndon visited Torreciudad, a shrine to the Virgin Mary associated with the controversial Catholic group Opus Dei, in Aragn, Spain. The sanctuary, built in the nineteen-seventies, sits on a cliff overlooking an inviting blue reservoir, in a remote area just south of the Pyrenees. Herndon and her husband, Mathew Dryhurst, had been on a short vacation in the mountains nearby. They were particularly taken with an exhibit of Virgin Mary iconography from around the world: a faceless, abstract stone carving from Cameroon; a pale, blue-eyed statuette from Ecuador; a Black Mary from Senegal, dressed in an ornate gown of blue and gold. Moving from art work to art work, the couple discussed Marys embedding. In machine learning, embeddings distill data down to concepts. They are what enable generative A.I. systems to process prompts such as Cubist painting of a tabby cat, wearing a hot-dog costume and eating a hot dog or country-club application, as a sestina. At Torreciudad, the sculptures and paintings on display all had aesthetic and material differences, yet there was something consistentineffable but essentialthat made the art works legible depictions of the same figure.

Around this time, Herndon and Dryhurst, who is also her primary collaborator, had been experimenting with the embedding of Holly Herndon in the data used to train text-to-image generators such as Dall-E and Stable Diffusion. Herndon, who is forty-three, has sea-glass-blue eyes, a round, pale face, and persimmon-colored hair; she tends to style it with bangs, a short bob in front, and a long braid in the back. The embedding of the Virgin Mary might be reduced to something involving her posture, gaze, and infant son; Herndons embedding is tied to her distinctive look. In 2021, she and Dryhurst began working on a series of computer-generated images, grouped under the title CLASSIFIED, that explored her embedding in an artificial neural network created by OpenAI. Though some of the art works are unsettling portraits of Herndonesque women rendered in the style of an oil painting, many are more playful: x|o40, which used the prompt A building that looks like Holly Herndon, shows a stately white structure with brick-red bangs, two porthole windows, and pursed pink lips; x | o 41 depicts a figure with buggy blue eyes and a red braid which could be fan art for The Simpsons. My identity in models is determined by aggregate cliches scraped from the web, Herndon recently tweeted. Im mostly a haircut!

Herndon is perhaps best known for her experimental electronic music, and for an art practice that spans the art world, academia, and the tech industry. She has performed and shown work at the Guggenheim, the Pompidou, and the Kunstverein in Hamburg; next year, she and Dryhurst have an exhibition at the Serpentine, in London, and will be part of a prestigious group show this spring in New York. (When asked if the group show was the kind that happened only biennially, Herndon declined to elaborate.) In recent years, she and Dryhurst have also fought for artists self-determination in the era of A.I. I always felt they were so far ahead of everybody else, Hans Ulrich Obrist, the artistic director of the Serpentine, said. They really think about what it does to the whole ecosystem: the artistic, the technical, the social, the economic aspects of these technologies.

Since 2020, Herndon and Dryhurst have been refining Holly+, a machine-learning model trained on Herndons voice. They refer to the model as a digital twin and a vocal deepfake, and see it as an experiment in decentralizing control of Herndons public identity. Ive never really fetishized my voice, Herndon told me. I always thought my voice was an input, like a signal input into a laptop. Holly+ can use a timbre-transfer machine-learning model to translate any audio filea chorus, a tuba, a screeching traininto Herndons voice. It can also be used in real time or be fed a score and lyrics: last year, Herndon gave a TED talk that opened with a recording of Holly+ singing an arrangement by Maria Arnal, a Catalan musician. It was a performance Herndon could never do. These beautiful, melismatic runsyou have to study that stuff for years, she said. (She also does not speak Catalan.) Several months later, Herndon released a track in which Holly+ covers Jolene, by Dolly Parton. Its glitchy, with oddly placed breaths and slurred phrases, and is weirdly compelling. A free version of Holly+ is available online. When I uploaded a clip of sea lions barking, it returned a grunting, stuttering, portentous motet.

Holly+ represents the future that Herndon and Dryhurst anticipate for music, art, and literature: a world of infinite media, in which anyone can adjust, adapt, or iterate on the work, talents, and traits of others. The two refer to the process of generating new media this way as spawningan act they distinguish from well-known forms of allusion such as sampling, pastiche, collage, and homage. When a d.j. samples an audio clip from another artist, the clip is copied, then recontextualized. Neural networks, on the other hand, dont reproduce their training data but represent its internal logicsomething like a style, a mood, or a vibe. Herndon uses the phrase identity playa pun of sorts on I.P.to describe the act of allowing other people to use her voice. What if people were performing through me, on tour? she said. Kind of like body swapping, or identity swapping. I think that sounds exciting. Decisions about what to do with Holly+ are made by a decentralized autonomous organizationa sort of coperative group of digital stewards. (Herndon retains a veto.) The musician Caroline Polachek told me, I see it as an inevitability that voice modelling will be outside of artists control, that people will eventually be able to use my voice with or without my consent. Holly specifically has woken up a lot of the art and music community to this window of time we have, to determine what we want to do with that.

Essentially a dress shoe, but you could run for your life in them.

Cartoon by Edward Frascino

In conversation, Dryhurst described Holly+ as an abstracted fork of Herndons identityin open-source-software development, forking is the act of copying source code and then changing it. Herndon alternated between calling it my voice and the voice. Its not like you dont have a relationship with that version of you, she told me. Its still an emotional connection, but its not you. Public identities already take on lives of their own, the couple noted; most of the publicly available images of Herndon, which CLASSIFIED drew from, are press photos. Years ago, while experimenting with machine-learning software, she and Dryhurst realized that all existing media could be used to train A.I. systems, an idea that now informs their art practice. As soon as something is machine-legible, its part of a training canon, Herndon told me. And thats very radicalizing.

We were sitting outside their bedroom in Berlin, in a white-walled apartment so spacious, high-ceilinged, and affordable that it felt almost like a slight. Their infant son, Link, played quietly with a babysitter in the living room. A large print by the artist Trevor Paglen, titled Tornado (Corpus: Spheres of Hell) Adversarially Evolved Hallucination, hung over the couch; it depicted a neural networks concept of a tornado. In the bedroom, previously Herndons music studio, large white acoustic panels hung from the walls and ceiling, framing a low, unmade bed and a small bookcaseMark Fisher, Michel Houellebecq, Baby-Led Weaning. A towering dieffenbachia plant, inherited from an elderly neighbor who had recently died, slouched against the doorframe. Dryhurst, who is thirty-nine, bald, and bespectacled, offered to demonstrate Holly+. See if it sounds the same with speech, Herndon, who was wearing white overalls, instructed. Dryhurst picked up a microphone, and chatted for a moment; Holly+, processing his voicehe has an English accentsounded drunk and a little congested. Its optimized for singing, Herndon said, laughing. Dryhurst sang a sequence of notes. After a tiny lag, Holly+ began to harmonize with him, and then the real Herndon joined in. The choral effect was pleasant, if chaotic. Shes definitely a better singer than I am, Herndon said.

Read more here:
Holly Herndon's Infinite Art - The New Yorker

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