JANUS: A Multi-modal Foundation Neural Sampler for Disordered Materials

Abstract: Many problems in disordered materials require sampling beyond fixed composition and volume, where coupled changes in atomic identities and structure create a prohibitively expensive discrete-continuous sampling problem. Here we introduce JANUS, a multimodal neural sampler that couples continuous and masked discrete diffusion through an equivariant graph neural network trained directly from energy evaluations, without pre-generated equilibrium data. In benchmark Ising and isobaric $ΔμNPT$ alloy systems, JANUS reproduces reference Monte Carlo equilibrium observables and recovers free energies and phase behavior with more than three orders of magnitude fewer energy evaluations. In multicomponent alloys, JANUS enables conditional steering toward prescribed chemical short-range order and enhanced bulk modulus and, when coupled to a LLM evolutionary agent, performs efficient inverse design for balanced optical and mechanical properties. In semiconductors like silicon and diamond, JANUS explores vacancies and dopants spanning 15 elements in grand-canonical $μVT$ ensembles, recovers established defects including the silicon $E$ centre, and identifies new candidate defect pairs and triplets for quantum engineering, including S-Ti in silicon and B-O-O in diamond, with deep in-gap states validated by hybrid-functional density functional theory. By unifying discrete site identities with continuous structural and volumetric relaxation, JANUS provides a foundation for thermodynamic sampling, characterization and inverse design of chemically disordered materials.

• The paper introduces JANUS, a multimodal generative model that jointly samples chemical occupations, atomic displacements, and cell volume across semi-grand-canonical and grand-canonical ensembles.
• JANUS learns directly from interatomic-potential evaluations through data-free fixed-point bootstrapping and achieves comparable thermodynamic accuracy with about 1,700-fold fewer energy evaluations than reference Monte Carlo.
• The framework estimates free energies, steers short-range order, discovers defect motifs, and supports alloy inverse design, while remaining limited to crystalline cubic systems and moderate supercell sizes.

JANUS is a multimodal neural sampler for crystalline disordered materials that jointly models discrete chemical occupations, continuous atomic displacements, and cell-volume fluctuations within a single generative framework trained directly on an interatomic potential, without pre-generated equilibrium data (2608.19116). The work addresses a persistent gap in neural thermodynamic sampling: existing approaches operate either on continuous $NVT$ ensembles or purely discrete lattice configurations, whereas realistic alloy and defect problems require sampling in ensembles such as isobaric semi-grand-canonical $\Delta\mu NPT$ and grand-canonical $\mu VT$, where chemical identity, structure, and volume are coupled.

Framework and training methodology

A configuration is represented as $(\mathbf a,\mathbf u,v)$: species tokens per lattice site (including a vacancy token), fractional displacements from reference sites, and log-volume. The target distribution combines the potential energy $U$ with pressure-volume and chemical-potential terms, so that only chemical-potential differences are physically meaningful and one species serves as the gauge reference. Generation couples two stochastic processes over $t\in[0,1]$: continuous channels evolve via a score-corrected SDE built on stochastic interpolants, while the fully masked species channel undergoes absorbing-state discrete diffusion in which each site is unmasked exactly once at a random time, drawing its identity from a learned denoising posterior conditioned on the full current state. All heads—velocities, scores, and categorical posteriors—are outputs of a single $E(3)$-equivariant PaiNN-style message-passing network that receives $(T,\Delta\mu)$ as inputs, amortizing one model across a continuous window of thermodynamic conditions.

Because no equilibrium data exists, JANUS is trained by data-free fixed-point bootstrapping. Each round generates terminal configurations under randomly sampled conditions; the interatomic potential labels them with forces, virial derivatives, and single-site heat-bath conditionals derived from substitution energies; intermediate states are formed by interpolation and re-masking; and regression plus soft cross-entropy losses update the heads. The authors prove in the Supplementary Information that the Boltzmann ensemble is a fixed point of this discrete iteration—fitting denoising posteriors by cross-entropy against heat-bath labels evaluated at self-generated terminals—which they state has not been studied before. They concede that uniqueness of this fixed point remains an open question. Physics-informed Gaussian priors for displacements ($\langle u^2\rangle\propto T$) and cell volume, parameterized entirely from the potential itself, shorten the prior-to-target transport.

Free energies and amortized observables

A key methodological contribution is that forward-backward path weights yield asymptotically unbiased partition-function estimates. The species-channel contribution to the weight is exact in configuration space—a sum of $N$ realized log-probabilities whose variance does not grow with grid refinement—while the continuous channels contribute closed-form Gaussian density ratios along the trajectory. Because the model is amortized, these weights give semi-grand-potential differences across $\Delta\mu NPT$0 from one training run. Composition-resolved free energies follow either through a Legendre-Fenchel transform of the semi-grand potential (used for the miscible Cu-Ni system) or through fixed-composition "ladder" sampling with Bennett acceptance ratio connecting adjacent rungs, which reaches inside miscibility gaps where importance weighting of a $\Delta\mu NPT$1-conditioned model would fail. For competing crystal structures, Frenkel-Ladd thermodynamic integration against Einstein references supplies absolute offsets.

Validation benchmarks

The discrete channel is first validated on the two-dimensional ferromagnetic Ising model on a $\Delta\mu NPT$2 lattice, where a single amortized model reproduces the magnetization landscape across the full temperature-field plane, including the paramagnetic-ferromagnetic transition at $\Delta\mu NPT$3. The full three-channel model is then tested on fcc Cu-Ni with 108 sites: JANUS reproduces composition response over 600–1200 K, mean displacement, atomic volume, pair distribution functions, and—without retraining—extrapolates the mixing free energy from $\Delta\mu NPT$4 to $\Delta\mu NPT$5 sites, enabled by size-intrinsic network parameters and exact scaling conventions for the collective volume channel.

The efficiency claim is strong: across Cu-Ni, Cu-Ag, and Ni-Cr, JANUS attains comparable thermodynamic accuracy with more than three orders of magnitude (~1700-fold) fewer energy evaluations than reference Monte Carlo. This amortization is what makes direct sampling with expensive universal machine learning interatomic potentials practical. On Cu-Ag, JANUS recovers the fcc miscibility gap; on Ni-Cr, separate lattice-specific samplers recover fcc-bcc competition and phase boundaries matching reference calculations.

Conditional steering and inverse design

Inference-time steering uses sequential Monte Carlo with exact Radon-Nikodym weights, so any proposal drift or tilted decoding leaves correctness unaffected—these choices control only variance. Applied to CrCoNi at 800 K, the base sampler already yields Warren-Cowley parameters ($\Delta\mu NPT$6, $\Delta\mu NPT$7) consistent with experimental Cr-Cr avoidance; reward tilting shifts the ensemble to prescribed targets ($\Delta\mu NPT$8, $\Delta\mu NPT$9) and enriches high-bulk-modulus configurations using Tweedie posterior-mean estimates and optional gradient guidance.

Coupled to an LLM evolutionary agent proposing chemical-potential conditions, JANUS performs multi-objective inverse design over Ag-Al-Au-Cu-Ni alloys, jointly optimizing bulk modulus, shear modulus, Pugh ratio, and visible-range reflectance (the latter via a GNN surrogate trained on 2048 DFT-computed dielectric spectra). The dominated hypervolume saturates within roughly 120 ensemble evaluations, yielding a broad Pareto front with interpretable elemental preferences: Al dominating reflectance, Ni stiffness, Ag contributing to both reflectance and ductility.

Defect discovery for quantum engineering

Amortized grand-canonical samplers spanning vacancies and 15 dopant elements explore silicon and diamond at 800 K in 216-site supercells, with concentration caps confining sampling to the dilute-defect regime. Motif enrichment ratios $\mu VT$0 relative to composition-matched random mixing rank pairs and connected triplets. The protocol spontaneously recovers established centers—the P-VAC E centre, As-VAC, and Sb-VAC in silicon, and the Ge-VAC colour centre in diamond—validating that enrichment reflects physical association. Hybrid-functional (HSE06) DFT with Freysoldt-Neugebauer-Van de Walle corrections identifies candidates with open-shell neutral ground states, finite neutral charge-stability windows, and deep occupied in-gap states: Fe-VAC ($\mu VT$1) and B-[O]-O ($\mu VT$2) in diamond—the latter with a notably deep stability window—and S-Ti ($\mu VT$3) and Ga-[Ga]-P ($\mu VT$4) in silicon. The authors state plainly that these DFT results do not establish quantum-engineering viability; zero-phonon-line energies, Huang-Rhys factors, and spin relaxation remain uncomputed.

Limitations and open questions

The paper concedes several boundaries. Scalability is demonstrated only at moderate supercell sizes, and quantitative phase-equilibrium results are limited to binary alloys; extending free-energy calculations to ternary and higher-order compositions requires improving the accuracy of the discrete diffusion channel, particularly near sharper first-order transitions. JANUS is restricted to cubic cells and crystalline basins—it does not treat melting, liquids, non-cubic lattices, off-lattice particle insertion/deletion, or dynamically changing coordination, so solid-liquid phase diagrams and interfacial problems remain out of reach. Whether the Boltzmann ensemble is the unique fixed point of the self-bootstrapped iteration is explicitly deferred to future work, as is formal treatment of tilted decoding variants beyond the outlined construction.

Conclusion

JANUS establishes, to the authors' knowledge, the first realization of joint discrete-continuous thermodynamic sampling in a machine learning model, unifying masked discrete diffusion over site identities with continuous diffusion over displacements and volume under flexible ensembles. Its combination of three-orders-of-magnitude energy-evaluation savings, amortization across temperatures and chemical potentials, system-size extrapolation, exact path-weight free-energy estimation, inference-time steering, and grand-canonical defect-space exploration constitutes a broadly applicable foundation for equilibrium thermodynamics, characterization, and inverse design of chemically disordered materials.