Tensor Moment Potentials in Materials & Imaging

• Tensor moment potentials are frameworks that represent energy and field transformations using invariant tensor descriptors, enabling accurate atomistic simulations and integral geometric inversions.
• They leverage systematic expansions and active learning to fit parameters against first-principles data, reducing computational cost while enhancing predictive fidelity.
• Extensions incorporate magnetic, dispersion, and many-body corrections, broadening application domains from materials discovery to advanced imaging and statistical inference.

A tensor moment potential is a framework for representing, analyzing, or learning potentials or transformations that rely fundamentally on the algebraic structure of tensor moments. Two distinct schools of research are associated with this term: (1) machine-learning interatomic potentials, especially the “Moment Tensor Potential” (MTP) and its extensions, used for atomistic simulations; and (2) mathematical and integral geometric constructions related to moment (integral) transforms of tensor fields. The following entry provides a rigorous exposition of the principles, mathematical frameworks, active learning strategies, and applications of tensor moment potentials, with particular focus on the MTP and its role in materials modeling, molecular simulation, and inversion theory.

1. Mathematical Foundations of Moment Tensor Potentials

At the core of the moment tensor potential (MTP) in atomistic modeling is the systematic expansion of the total energy as a sum over atomic contributions, each expressed as a linear combination of invariant basis functions constructed from projections of local geometric and, if relevant, additional physical fields (e.g., spin):

$$E(x) = \sum_{i=1}^N V(r_i),\quad V(r_i) = \sum_{j=1}^m \theta_j B_j(r_i)$$

where:

• $N$ is the number of atoms,
• $V(r_i)$ is the atomic energy associated with atom $i$ and its neighborhood $r_i$,
• $B_j$ are basis functions capturing geometric correlations, constructed from moment tensor descriptors,
• $\theta_j$ are parameters fitted to first-principles (DFT) data.

The moment tensor descriptors have the form:

$$M_{\mu,\nu}(n_i) = \sum_{j} f_\mu(|r_{ij}|, z_i, z_j) [r_{ij}^{\otimes \nu}]$$

with:

• $f_\mu$ a species-dependent radial function (typically expanded using Chebyshev polynomials and a smooth cutoff),
• $r_{ij}$ the relative position vector from atom $i$ to $j$,
• $\otimes \nu$ denotes the tensor product of rank $\nu$,
• $z_i$, $z_j$ denote atomic species.

The basis functions $B_j(n_i)$ are formed from scalar contractions of such moment tensors, ensuring invariance with respect to rotation, translation, and particle permutation. The expansion is systematically improvable by increasing the level (body-order and angular completeness) parameter, and the energy is differentiable with respect to both positions (giving forces) and cell parameters (giving stresses).

For classical moment tensor transforms in integral geometry, the object of interest is a symmetric $m$-tensor field $f$, studied through its weighted line-integrals ("moment transforms"):

$$(I^q f)(x, \xi) = \int_{-\infty}^{\infty} t^q \langle f(x + t\xi), \xi^{\otimes m} \rangle \, dt$$

where $\langle \cdot, \cdot \rangle$ denotes contraction.

These transforms are intimately connected to the decomposition of tensor fields into solenoidal and potential (gauge) parts, and underpin inversion and range characterization problems in medical and geophysical imaging.

2. Training and Active Learning Strategies

The parameterization of MTPs is achieved by regression to first-principles data, minimizing a composite loss that balances errors in energy, forces, and stress:

$$\mathcal{L} = \sum_{x \in X_{TS}} \left[ C_E^2 (E(x) - E^{\text{DFT}}(x))^2 + C_f^2 \sum_j |f_j(x) - f_j^{\text{DFT}}(x)|^2 + C_s^2 (\sigma(x) - \sigma^{\text{DFT}}(x))^2 \right]$$

Efficient coverage of phase space is achieved through "active learning on-the-fly." Here, during MD or enhanced sampling:

• Each new configuration $x^*$ is assigned an extrapolation grade $\gamma(x^*)$ based on the sensitivity of $E(x^*)$ to fitted parameters,
• Configurations with high $\gamma(x^*)$ (evaluated using D-optimality or the maxvol algorithm) are flagged for DFT evaluation and added to the training set,
• This process ensures that rare-event regions (e.g., near diffusion barriers, phase transitions) are sampled and parametrized with high fidelity.

The active learning protocol is essential for the robust extension of MTPs to new domains, including high-temperature dynamics and non-equilibrium or disordered phases. The number of required DFT evaluations is minimized (typically hundreds–thousands, depending on system complexity and required accuracy), and the method has been shown to outperform conventional fixed database approaches, particularly in sampling saddle-point regions critical for rare-event-driven phenomena like diffusion.

3. Extensions: Magnetism, Dispersion, and High-Order Interactions

Magnetic Moment Tensor Potentials (mMTP)

For systems where magnetic degrees of freedom play a structural or dynamical role, mMTPs extend the neighborhood descriptors to incorporate collinear spin variables $s_i$ (and possibly higher moments), with the total energy written as:

$$E(\{r\},\{s\}) = \sum_{i} V(n_i),\quad n_i = \{ (r_{ij}, s_i, s_j),\, \forall j \}$$

The basis functions are defined over both geometric and magnetic variables, and fitting is performed including magnetic force data (energy derivatives with respect to spins), allowing for faithful reproduction of ab-initio energetics across magnetic states (FM, AFM, PM). Constrained DFT sampling and AL strategies are used to explore both equilibrium and nonequilibrium magnetic configurations.

Explicit Dispersion and Many-Body Corrections

To account for long-range van der Waals/dispersion missing from local descriptors (especially with short cutoffs), explicit corrections (e.g., D2 or D3) are added:

$$E^{\text{Total}} = E^{\text{MTP}} + E^{\text{Disp}}$$

$$E^{\text{Disp}} = -s_6 \sum_{i,j} \frac{C_6^{ij}}{r_{ij}^6} f_{\text{damp}}(r_{ij})$$

This is crucial in molecular crystals and liquids (e.g., toluene, CCl₄) where local potential truncation leads to underbinding. For metallic and ionic systems, increasing the cutoff plays a complementary role, but explicit dispersion corrections improve accuracy in systems with anisotropic and orientation-dependent interactions.

Advanced Tensor Decomposition and Computational Optimizations

The computational cost of constructing invariant basis functions can be significant with increasing tensor order. Genetic optimization algorithms have been developed to minimize the number of independent tensor contractions and intermediates by searching over contraction trees, yielding up to an order-of-magnitude improvement in simulation efficiency, particularly for intricate or high-angular-momentum basis sets.

4. Applications: Materials Modeling, Diffusion, Phase Diagrams, and Beyond

MTPs and their extensions have been applied to a diverse array of physical systems including:

• Vacancy diffusion in metals and semiconductors: Extraction of vacancy diffusion coefficients and migration energies in Al, Mo, Si using the Einstein–Smoluchowski relation and explicit modeling of rare events (Novoselov et al., 2018, Zongo et al., 2023).
• Thermophysical properties of molten salts: Accurate calculations of diffusion coefficients, densities, viscosities, and thermal conductivities for FLiBe and FLiNaK, with deviations matching experimental uncertainty and efficiency surpassing neural network potentials (Attarian et al., 2023, Nikita et al., 28 Feb 2024).
• Structure prediction and phase diagrams: Active-learning-driven crystal structure prediction for molecular crystals (benzene, glycine), robust evaluation of phase boundaries and eutectic points in metallic alloys (Ag–Cu), and accurate interface, defect, and stacking fault energies (Rybin et al., 4 Oct 2024, Nitol et al., 25 Aug 2025).
• Phonon and thermal transport: High-accuracy modeling of lattice thermal conductivity in a/b-Ga₂O₃ and assessments of many-body heat current operators, with corrections to the virial and heat flux calculation shown to impact computed $\kappa_L$ by up to 64% (Rybin et al., 29 Feb 2024, Tai et al., 2 Nov 2024).
• Amorphous and disordered systems: Synergistic use of activation relaxation methods and MTPs yields defect-free, realistic amorphous silicon models that resolve the continuous random network hypothesis (Zongo et al., 17 Jan 2025).
• Quantum information theory: Tensor moment formalism is utilized to exactly compute high-order moments in local random quantum circuits using tensor networks, outperforming Monte Carlo estimators and enabling the paper of anticoncentration and unitary designs (Braccia et al., 4 Mar 2024).
• Machine learning in statistical inference: The development of implicit computation methods for tensor moments of Gaussian mixtures establishes efficient moment-matching algorithms, circumventing the curse of dimensionality and connecting moment tensors to homogeneous polynomials and Bell polynomials (Pereira et al., 2022).

5. Mathematical Theory: Tensor Moment Transforms in Integral Geometry

Beyond the context of molecular modeling, tensor moment potentials also arise in the analysis of transforms over symmetric tensor fields. The "integral moment transform" generalizes classical Radon transforms by including weighted (moment) integrals:

$$(I^q f)(x, \xi) = \int_{-\infty}^{\infty} t^q \langle f(x + t\xi), \xi^{\otimes m} \rangle \, dt$$

Key results include:

• Generalized Solenoidal–Potential Decomposition: Any smooth symmetric $m$-tensor field $f$ can be split uniquely into $k$-solenoidal ($\delta^k f = 0$) and $k$-potential ($f = d^k v$) components.
• Injectivity and Kernel: The (k+1)-moment transform is injective on (k+1)-solenoidal tensors; its kernel comprises the (k+1)-potential fields.
• Range Characterization: The necessary and sufficient conditions for data to be in the range of combined moment transforms are expressed via the vanishing of iterated John operators applied to constructed functions $\psi^\ell$, with explicit symmetry conditions.

This mathematical structure underpins inversion approaches in photoacoustic tomography, seismic imaging, and other applied fields, dictating precisely what part of the "physical field" can be recovered from moment-weighted integral data, and providing differential constraints (John's equations) for data consistency (Mishra et al., 2020).

6. Impact, Challenges, and Future Prospects

Tensor moment potentials, and especially MTPs, have enabled a paradigm shift in atomistic and statistical modeling:

• They offer precision close to ab initio methods (e.g., DFT) with orders-of-magnitude speedup, closing the gap between empirical potentials and quantum accuracy.
• The modular, systematically improvable descriptor framework is extendable to new physics: explicit magnetism (mMTP), tensorial properties, explicit environmental and long-range effects.

Challenges and active research include:

• Extension to multicomponent and multi-field systems (e.g., charge, magnetization, polarizability).
• Incorporating explicit quantum effects efficiently in classical or mixed quantum–classical simulations.
• Systematic reduction of the descriptor space via machine learning and optimization, especially for complex materials or large molecules.
• Theoretical analysis of identifiability, invertibility, and domain of applicability, especially in the context of active learning and extrapolation grading.

A plausible implication is that tensor moment potentials may provide a unifying mathematical and computational framework, adaptable across physics-based atomistic simulation, statistical inference, and inverse problems involving tensor fields and integral geometry, with broad implications for materials discovery, chemistry, and computational mathematics.