MADField: Multi-fidelity Amortized Density Field for Adsorption in Nanoporous Materials

Abstract: High-throughput computational screening of nanoporous materials for gas storage and separation requires fast and accurate characterization of adsorption equilibrium. Particle-based grand canonical Monte Carlo (GCMC) and density-based classical density functional theory (cDFT) provide simulation-based estimates of gas uptake and adsorbate density fields, but their speed-accuracy tradeoff remains insufficient for large-scale screening. In this work, we address this gap with Multi-fidelity Amortized Density Field for Adsorption in Nanoporous Materials (MADField), which reframes adsorption prediction as equilibrium density-field estimation. MADField learns from two complementary fidelities, combining broad and scalable cDFT density supervision with higher-fidelity GCMC density labels, and recovers gas uptake by integrating the predicted density field. MADField improves uptake accuracy over the strongest baselines by 6.0x for cDFT and 15.4x for GCMC, and its predicted fields accelerate cDFT solvers with 2.0x fewer iteration steps while recovering 42 percent of cases that fail under standard settings. Finally, we evaluate MADField for conventional CH4 working capacity screening on the 270k-structure ARC-MOF database. Within this space of extremely rare high-capacity targets, 167 in total, the model achieves 56x higher average precision than the strongest baseline and accelerates inference by five orders of magnitude compared to GCMC. By prioritizing the MADField rankings, selecting the top 1.7 percent of candidates recovers 95 percent of all targets, while the top 6 percent ensures 100 percent recall.

• The paper introduces a novel method to predict physically scaled 3D adsorbate density fields, enabling direct computation of scalar uptake and spatial adsorption structure.
• It employs a multi-fidelity training approach that pre-trains on cDFT data and fine-tunes on GCMC results, achieving state-of-the-art accuracy across diverse material classes.
• MADField dramatically accelerates high-throughput screening by reducing errors and doubling solver convergence rates, facilitating efficient materials discovery.

MADField: Multi-fidelity Amortized Density Field for Adsorption in Nanoporous Materials

Problem Setting and Motivation

Characterization of adsorption equilibrium in nanoporous materials, especially MOFs, is a key bottleneck in high-throughput computational screening for gas storage and separation. Two main physical modeling paradigms are prevalent: particle-based GCMC, which is considered the gold standard but is computationally prohibitive for large-scale screening, and density-based classical density functional theory (cDFT), which is more tractable but suffers from functional approximations and convergence limitations. Existing ML approaches have focused largely on scalar uptake regression, typically using global or local features but discard spatial adsorption structure and show limited accuracy, especially when extrapolating. Furthermore, recent field prediction models are fundamentally normalized or auxiliary-data-dependent, incapable of recovering physically scaled density fields critical for uptake estimation and solver initialization.

MADField Architecture and Methodology

MADField reframes adsorption prediction as a task of equilibrium density-field modeling, predicting unnormalized 3D adsorbate density fields in physical units for a given porous framework, adsorbate species, temperature, and pressure. The predicted density field enables direct computation of scalar uptake by spatial integration, captures spatial adsorption phenomena, and is compatible with numerical solver initialization for cDFT.

The approach leverages multi-fidelity learning:

• Coarse fidelity: Pre-train on large-scale, physically approximated (but fast) cDFT-generated density fields across nine non-polar adsorbates and a wide structural manifold, providing geometric and thermodynamic coverage.
• Fine fidelity: Fine-tune using sparse but high-precision GCMC-derived density fields, leveraging the cDFT prior for data efficiency and to bridge the remaining functional fidelity gap.

MADField's backbone is a U-shaped 3D Swin Transformer operating on periodic volumetric grids, in conjunction with a conditioning protocol based on the Boltzmann reference density and molecule-specific physical descriptors (PC-SAFT or TraPPE-type parameters). The model employs a log-density residual parameterization to focus learning capacity on many-body corrections beyond ideal-gas behavior.

Multi-fidelity Density Field Supervision and Training

Training utilizes supervised objectives at the density field level, blending log-residual, relative error, and auxiliary losses over both the physically accessible pore region and high-density adsorption sites. The uptake objective is formulated by integrating the predicted field and matching it against ground-truth loadings. The pre-training stage (MADField-cDFT) incorporates an internal bootstrapping protocol where model predictions are used to initialize failed solver runs, thereby expanding converged cDFT supervision without polluting ground truth with neural pseudo-labels.

Fine-tuning to GCMC data (MADField-GCMC) is realized with LoRA-adapted adapters, efficiently aligning the model to high-fidelity simulation references with minimal additional parameters.

Numerical Results and Comparative Evaluation

Quantitative benchmarks on both in-distribution (ID; MOF) and out-of-distribution (OOD; amorphous carbon, PIM, HCP, Kerogen) material classes reveal that MADField achieves order-of-magnitude reductions in mean absolute error relative to the strongest deep learning baselines, for both physically scaled density fields and scalar uptake. For uptake prediction on MOFs, MADField attains 0.82 cm$^3$/g (cDFT) and 0.58 cm$^3$/g (GCMC) MAE—improving over the best competitor by factors of $6.0\times$ and $15.4\times$, respectively. On OOD disordered materials, MADField maintains sub-1 cm$^3$/g error across challenging cases with up to 18,000 atoms/unit cell, whereas baseline performance typically degrades by an order of magnitude.

(Figure 1)

Figure 1: CH$_4$ working-capacity screening result on the ARC-MOF database, highlighting MADField's recall and precision in identifying rare high-capacity targets.

MADField's predicted density fields enable reconstruction of spatial adsorption structure and serve as effective initializations to accelerate cDFT solver convergence, doubling the convergence rate and salvaging 42% of cases that fail under standard initialization.

On large-scale materials discovery workflows, MADField enables database-scale screening of over $2.7\times 10^5$ MOFs for CH$_4$ working capacity. When ranking candidates by predicted working capacity, it recovers 99.9% of the rare high-capacity frameworks found by exhaustive GCMC, while accelerating the workflow by five orders of magnitude relative to direct GCMC and vastly outperforming all other ML baselines in average precision (MADField AP = 0.557 vs. Uni-MOF AP = 0.010, a $56\times$ improvement).

Implications, Limitations, and Future Directions

MADField introduces a unification of scalar property prediction and spatial adsorption modeling, making physically scaled density fields the fundamental surrogate. This enables both interpretability (e.g., spatial localization of adsorbates), enhanced generalization to new structural classes, and direct utility as a drop-in acceleration for physics-based solvers—bridging the gap between high-throughput screening and first-principles methods.

The principal limitation is that pre-training relies on the accuracy of underlying cDFT functionals; however, empirical results indicate that subsequent GCMC fine-tuning largely closes the fidelity gap, and the benchmarked adsorbates are well treated by PC-SAFT. Extension to mixture adsorption, multicomponent competitive effects, and polar/associative adsorbates remain open directions, as does integration with fully differentiable simulation pipelines and uncertainty quantification for active learning.

Conclusion

MADField demonstrates that equilibrium density-field prediction with multi-fidelity neural operators provides a highly scalable, accurate, and generalizable foundation for adsorption simulation and screening in nanoporous materials. The model's explicit density formulation and strong physical conditioning enable new capabilities—in spatial structure resolution, solver acceleration, and database-scale identification of high-performance frameworks—that are both practical and of theoretical interest to materials informatics and molecular simulation communities.