SCALE: Simulation-Calibrated Amortized Learning for Energy Materials (A hybrid architecture connecting deterministic modeling, real-world data, and transformer-scale inference for accelerated energy-materials discovery)

Kuan Huang; Bo Bai

cs.LGSep 3, 2026

SCALE: Simulation-Calibrated Amortized Learning for Energy Materials (A hybrid architecture connecting deterministic modeling, real-world data, and transformer-scale inference for accelerated energy-materials discovery)

Authors: Kuan Huang, Bo Bai

Organizations: Alpha Ladder Particle Pte. Ltd., 168 Robinson Road, #26-02, Singapore 068912

Abstract

Energy systems face converging pressures for security, affordability, resilience, and sustainability, creating a need for faster discovery of deployable energy materials. Here we introduce SCALE (Simulation-Calibrated Amortized Learning for Energy Materials), a physics-grounded, real-world-data-calibrated learning architecture that connects deterministic scientific operators, experimental calibration, expanded calibrated label generation, and transformer-scale inference. SCALE converts selected high-cost mechanistic computation and measured evidence into reusable models for rapid screening, ranking, inverse design, and active learning. We formulate the framework, identify ten method-based application regimes, and demonstrate SCALE for solid-state metal-hydride hydrogen-storage capacity prediction. In this implementation, a hydride phase-equilibrium capacity operator is calibrated against 381 measured ML-HydPARK capacity anchors and used to generate 5,000 candidate-condition-prototype teacher labels. A crystallographically anchored periodic-graph representation preserves atomic sites, periodic neighbor relationships, and local metal environments absent from formula-only encodings. An edge-biased graph transformer with 2.90 million parameters reproduces calibrated teacher labels with five-fold surrogate fidelity of MAE 0.0582 wt% H2, RMSE 0.0833 wt% H2, R2 = 0.9927, and Pearson r = 0.9963. Post hoc attention analysis suggests that SCALE learns chemically organized element groupings and metal-metal relationships consistent with established hydride chemistry, without chemistry-group labels as supervision. Once trained, SCALE shifts million-candidate evaluation from repeated deterministic workflow execution to batched learned inference, reducing per-candidate screening cost by approximately 10^7-10^8 while retaining links to simulation and experimental evidence.

Figures & tables

Panel of Figure 1 — Figure 1: SCALE separates calibrated evidence generation from deployment inference. Deterministic operators and experiments are used offline to construct calibrated teacher labels for training; after training, large candidate permutations are evaluated by the learned model, and active-learning outputs return to experimental validation.

Category	Representative deterministic tools/workflows	Representative energy materials	Representative studies and SCALE role	Deterministic workload
Electronic-structure screening	VASP, Quantum ESPRESSO, CP2K, CASTEP, ABINIT, GPAW, atomate, AiiDA	Battery cathodes/anodes, solid electrolytes, perovskite PV absorbers, electrocatalysts, thermoelectrics, permanent magnets, hydrides, corrosion alloys, fusion alloys	Materials Project, AFLOW, and OQMD demonstrate high-throughput first-principles screening [ 9 , 10 , 11 ] ; atomate automates first-principles workflows [ 12 ] . SCALE can calibrate electronic-structure outputs against measured performance and distill them into composition/structure transformers.	Hours–days per structure; $10^{4}$ – $10^{7}$ core-hours per campaign.
Finite-temperature atomistic transport	AIMD, NEB, CP2K, VASP-AIMD, Quantum ESPRESSO CP, VTST	Solid electrolytes, hydride diffusion pathways, perovskite ion migration, catalytic surfaces, radiation-defect migration	First-principles dynamics and migration-barrier workflows provide mechanistic transport labels. SCALE can combine these labels with measured conductivity, diffusion, or degradation data for rapid transport screening.	Hours–days per path; $10^{3}$ – $10^{6}$ core-hours per campaign.
Classical and reactive molecular dynamics	LAMMPS, GROMACS, OpenMM, AMBER, CHARMM, NAMD, ReaxFF	Polymer electrolytes, PEM/AEM membranes, MOFs, CO 2 sorbents, SEI/CEI layers, catalyst-ionomer films, phase-change materials, thermal-interface materials	Molecular simulation has enabled large adsorption and transport studies in porous materials [ 31 ] . SCALE can convert hydration, uptake, transport, swelling, morphology, and degradation simulations into calibrated transformer labels.	Hours–days per system; $10^{3}$ – $10^{6}$ core-hours per campaign.
Machine-learned interatomic potentials	M3GNet, CHGNet, MatterSim-style potentials, DeePMD-kit, MACE, NequIP, GAP	Batteries, hydrides, high-entropy alloys, catalysts, perovskites, MOFs, fusion alloys, thermoelectrics	M3GNet and CHGNet show broad learned-potential transfer [ 16 , 17 ] ; large atomistic models further broaden this direction [ 18 ] . SCALE can place experimental calibration above ML-potential simulations.	High training cost; low–moderate inference cost per trajectory.
Thermodynamics and phase equilibria	Thermo-Calc, Pandat, FactSage, OpenCalphad, pycalphad	Hydrides, high-entropy alloys, molten salts, battery alloys, fusion blankets, nuclear alloys, thermoelectrics, phase-change alloys	pycalphad provides open CALPHAD infrastructure [ 32 ] . SCALE can learn calibrated phase-stability, phase-fraction, and plateau-pressure response surfaces.	Seconds–minutes per query after database assessment.
Reaction kinetics and kinetic Monte Carlo	Cantera, CHEMKIN, COPASI, Zacros, custom KMC solvers	Polymerization, electrolyte aging, CO 2 reduction, OER/HER/ORR catalysts, ammonia cracking, liquid organic hydrogen carriers	Zacros supports lattice KMC for catalytic surfaces [ 33 ] . SCALE can convert mechanism-based simulations and reaction experiments into fast reaction-outcome models.	Seconds–hours per ODE case; $10^{2}$ – $10^{5}$ core-hours for KMC.

Table 1: Method-based benchmark regimes for SCALE. Compute-density values are order-of-magnitude planning estimates if only deterministic models are used directly for screening before transformer amortization; actual cost depends on system size, convergence criteria, hardware, licensing, and campaign design.

Step	Stage	Input	Output	Tools and method	Computational burden
1	Parent-alloy definition	Formula, material class, $T$ , and $P_{\mathrm{H_{2}}}$	Parsed composition, molar mass, and condition record	pymatgen and validated composition parsing [ 45 ]	Low; linear in the number of elements, $O(E)$ .
2	Parent-structure resolution	Parsed formula and parent-only structure sources	Exact parent CIF or ordered parent approximant	Materials Project or OQMD retrieval; deterministic supercell and sublattice substitution [ 9 , 11 , 45 ]	Low–moderate; API/cache dominated for exact structures, increasing with approximant size.
3	Parent quantum relaxation	Parent CIF, pseudopotentials, $k$ -mesh, cutoffs, and smearing	Relaxed or plateau-accepted parent geometry and energy	Quantum ESPRESSO variable-cell relaxation with PBE pseudopotentials [ 46 ]	High; approximately $C_{\mathrm{vc}}(N,K,I)\sim N^{2.5\text{--}3}KI_{\mathrm{scf}}I_{\mathrm{geo}}$ .
4	Parent-structure analysis	Relaxed parent geometry	Distances, symmetry/prototype, and candidate interstitial regions	pymatgen , spglib , periodic distance analysis, and Voronoi/grid void search [ 45 , 47 ]	Low–moderate; pair analysis $O(N^{2})$ , with void-search cost set by grid density.
5	Hydrogen-loading hypotheses	Candidate interstices, formula units, and collision constraints	Parent+H structures across discrete loading states	Deterministic H-site placement, loading enumeration, and structure/input writers	Moderate; scales with candidate sites and loading branches, approximately $O(BN)$ .
6	Fast quantum screening	Parent+H candidates and parent/H 2 references	Ranked loading candidates	Fixed-cell or short-relaxation Quantum ESPRESSO screening, energy parsing, and formation-energy ranking	High, but screening grade; approximately $B\,C_{\mathrm{scf}}(N_{H},K,I)$ .

Table 2: Hydride phase-equilibrium capacity operator used in the metal-hydride implementation. The workflow begins with the hydrogen-free parent alloy, enumerates hydrogen-loading hypotheses without using measured capacity or known hydride stoichiometry, and promotes selected branches to full first-principles confirmation. Computational-burden expressions describe deterministic work rather than hardware-specific wall time. Because the present implementation evaluates the SCALE architecture rather than establishing a definitive metal-hydride phase diagram, Step 7 was benchmarked on representative candidate branches to quantify the computational burden of terminal variable-cell confirmation but was not executed exhaustively across the expanded corpus. Note:

N

is the number of atoms in the quantum cell,

K

is the number of irreducible

k

-points,

I_{\mathrm{scf}}

is the number of electronic iterations per geometry step,

I_{\mathrm{geo}}

is the number of ionic or cell-relaxation steps,

B

is the hydrogen-loading ladder size, and

m

is the number of branches promoted to full relaxation. Heavy elements, semicore states, metallic smearing, low symmetry, and large ordered approximants can substantially increase the computational prefactor.

Model	Validation split	$n$	MAE	RMSE	$R^{2}$	Pearson $r$
Hydride capacity operator	all rows	381	0.623	0.788	0.274	0.823
Ridge calibration	random 5-fold	381	0.291	0.450	0.763	0.875
Histogram-gradient boosting	random 5-fold	381	0.230	0.371	0.839	0.916
Random forest	random 5-fold	381	0.234	0.367	0.843	0.918
Extremely randomized trees	random 5-fold	381	0.219	0.380	0.831	0.912
Extremely randomized trees	material-class grouped	381	0.405	0.610	0.565	0.761

Table 3: Experimental calibration accuracy against ML-HydPARK measured H 2 wt%. The selected calibration uses deployable chemistry/condition descriptors and deterministic-operator outputs; raw operator performance directly compares uncalibrated simulation-derived capacity to experiment.

Split	$n$	MAE	RMSE	$R^{2}$	Pearson $r$	Bias
Overall OOF	5000	0.0582	0.0833	0.9927	0.9963	$+0.0022$
Fold 1	1000	0.0593	0.0901	0.9919	0.9960	$+0.0080$
Fold 2	1000	0.0591	0.0849	0.9929	0.9964	$-0.0017$
Fold 3	1000	0.0582	0.0825	0.9920	0.9960	$+0.0021$
Fold 4	1000	0.0589	0.0810	0.9932	0.9966	$-0.0026$
Fold 5	1000	0.0556	0.0773	0.9934	0.9967	$+0.0054$

Table 4: Overall and fold-level graph-transformer surrogate fidelity against calibrated metal-hydride H 2 wt% teacher labels. The overall row reports metrics calculated from all 5,000 out-of-fold predictions; fold rows demonstrate consistency across independently trained models.

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Active learning (AL) has emerged as a powerful paradigm for accelerating materials discovery by iteratively steering experiments toward promising candidates, reducing the number of costly synthesis-and-characterization cycles needed to identify optimal materials. However, current AL relies predominantly on Gaussian Process (GP) and Random Forest (RF) surrogates, which suffer from complementary limitations: GP underfits complex composition-property landscapes due to rigid kernel assumptions, while RF produces unreliable heuristic uncertainty estimates in small-data regimes. This small-data challenge is pervasive in materials science, making reliable surrogate modeling extremely difficult with models trained from scratch on each new dataset. Here we propose In-Context Active Learning (ICAL), which addresses this bottleneck by replacing conventional surrogates with TabPFN, a transformer-based foundation model (FM) pre-trained on millions of synthetic regression tasks to meta-learn a universal prior over tabular data, upon which TabPFN performs principled Bayesian inference in a single forward pass without dataset-specific retraining, delivering strong small-data regression performance and well-calibrated predictive uncertainty (required for effective AL). We benchmark ICAL against GP and RF across 10 materials datasets and TabPFN wins on 8 out of 10 datasets, achieving a mean saving of 52% in extra evaluations relative to GP and 29.77% relative to RF. Cross-validation analysis confirms that TabPFN's advantage stems from superior uncertainty calibration, achieving the lowest Negative Log-Likelihood and Area Under the Sparsification Error curve among all surrogates. These results demonstrate that pre-trained FMs can serve as effective surrogates for active learning, enabling data-efficient discovery across diverse materials systems and small-data experimental sciences.

Abstract

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