Mathematical Invariant-Enabled Topological Neural Networks for Molecular and Materials Property Prediction
Authors: Yiming Ren, Xiang Liu, Mustafa Hajij, Pietro Liò, Guo-Wei Wei
Organizations: Department of Mathematics, University of Georgia, Athens, GA 30602, USA. · Department of Data Science, University of San Francisco, CA 94117, USA. · Department of Computer Science and Technology, Cambridge University, Cambridge, United Kingdom. · Department of Biochemistry and Molecular Biology, University of Georgia, Athens, GA 30602, USA. · School of Computing, University of Georgia, Athens, GA 30602, USA.
Existing molecular and materials learning approaches often rely on a limited set of structural representations, which may capture only selected aspects of complex three-dimensional structure. Here, we introduce mathematical invariant-enabled topological neural networks (MITNNs), a framework that represents complex structures through multiple complementary mathematical views and integrates them with topological neural architectures. MITNNs combine multiscale invariants from topology, spectral theory, commutative algebra, differential geometry, and discrete curvature, capturing complementary structural information from the same system. Systematic invariant-subset, architecture-subset, and ensemble analyses show that predictive performance depends on how mathematical representations and neural architectures are paired, with selected combinations outperforming individual models and the aggregation of all available components. Across protein-ligand binding, metal-organic framework properties, mutation-induced protein solubility, and molecular toxicity prediction, MITNN consistently outperforms existing methods. These results establish MITNN as a mathematically multimodal framework for scientific machine learning.
Figures & tables
Figure 1 : Overview of the MITNN framework. (A) Representative molecular and materials systems, including (A1) protein–ligand complexes, (A2) metal–organic frameworks, (A3) wild-type and mutant protein structures, and (A4) small molecules. (B) Mathematical domains, including (B1) bipartite graphs, (B2) Vietoris–Rips and Alpha simplicial complexes, and (B3) differentiable manifolds. (C) Mathematical invariant families, including (C1) commutative algebra, (C2) persistent homology, (C3) persistent Laplacian, (C4) element interactive curvature, and (C5) Forman persistent Ricci curvature. (D) Neural architectures, including the topological neural architectures (D1) simplicial neural networks and (D2) copresheaf transformer neural networks, and the classic neural architectures (D3) artificial neural networks and (D4) convolutional neural networks. (E) Consensus prediction obtained by aggregating invariant–architecture base models.
Figure 2 : Overall predictive performance of MITNN across molecular and materials benchmark applications. Results are shown for both MITNN consensus variants. (A) Protein–ligand binding affinity prediction on CASF-2016, evaluated by PCC and SD and compared with representative scoring functions and topology-based models [ 31 , 14 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ] . (B) O 2 and N 2 uptake prediction in MOFs, evaluated by R2 , MAE, and RMSE and compared with representative descriptor-based, topology-based, and transformer-based approaches [ 39 , 40 , 41 , 42 , 43 ] . (C) Mutation-induced protein solubility prediction on PON-Sol2, evaluated by CPR and GC2 and compared with existing solubility predictors [ 44 , 45 ] . (D) Quantitative LD 50 toxicity prediction, evaluated by PCC2 and RMSE and compared with representative traditional machine-learning, multitask-learning, and molecular-descriptor-based methods [ 46 , 47 , 48 , 49 , 50 , 51 ] .
Figure 3 : Complementarity among mathematical invariant families in MITNN. (A) Exhaustive invariant-subset analysis for CASF-2016 and LD 50 . The incidence matrix indicates the invariant families included in each subset. For each neural-network architecture, predictions from the corresponding base models were averaged to form the subset prediction. The heatmaps report PCC for CASF-2016 and PCC2 for LD 50 across four architectures. (B) Performance versus invariant-subset size for CASF-2016, MOF O 2 uptake, PON-Sol2, and LD 50 . Each point denotes one invariant subset under a fixed architecture; solid lines show the mean over subsets of the same size across the four architectures, with shaded regions indicating variability. CASF-2016, PON-Sol2, and LD 50 use PH, PL, CA, FPRC, and EIC, while MOF O 2 uptake uses PH, PL, CA, and FPRC.
Figure 4 : Architectural complementarity and selective ensemble composition in MITNN. (A) Architecture-subset analysis with the mathematical invariant held fixed. For each invariant, predictions from the corresponding invariant–architecture base models within a given architecture subset were averaged to form the subset prediction. Points show the mean performance over all architecture subsets of the same cardinality, and shaded regions indicate the corresponding variability. (B) Ensemble-performance landscapes over the complete invariant–architecture base-model pool for CASF-2016, PON-Sol2, LD 50 , and MOF O 2 uptake. For each ensemble size, the performance distribution was obtained over all candidate base-model subsets of that cardinality. Solid and dashed curves denote the median and best performance at each ensemble size, respectively, and shaded regions indicate the 25–75 and 10–90 percentile intervals. Stars mark MITNN best , corresponding to the global maximum of each ensemble landscape, whereas hollow squares mark MITNN all at the complete-pool endpoints. (C) Inclusion frequency of each invariant–architecture base model among the top 100 performing ensembles for each application.
Figure 5 : Schematic comparison of four filtration-based multiscale mathematical invariants obtained from the same Vietoris–Rips filtration of a 14-point two-loop point cloud. (A) Representative Vietoris–Rips complexes at increasing filtration parameter ϵ . (B) Persistence intervals for H0 and H1 . (C) Evolution of the nonharmonic spectrum of the 0 -Laplacian L0 , illustrated by the smallest and largest positive eigenvalues λmin+ and λmax+ , the mean positive eigenvalue λˉ , and the nonharmonic spectral range. (D) Facet-persistence bars P0 , P1 , and P2 associated with 0 -, 1 -, and 2 -dimensional facets. (E) Signed mean Forman curvature mean(Fp) for p=0,1,2 .
Molecular message-passing neural networks commonly propagate chemically diverse interactions through a single graph, which may mix interaction-specific signals and require deep propagation to capture long-range effects. We introduce the Multi-level, Multi-color Graph Neural Network (MMGNN), a hierarchical framework that decomposes a molecular graph into overlapping atom-type-pair-specific subgraphs while preserving atom-level resolution. MMGNN-2D constructs chemical-colored subgraphs from covalent connectivity, whereas MMGNN-3D constructs geometric-colored subgraphs from spatial proximity and augments their edges with distance, angular, and torsional descriptors. Both variants apply a shared communicative message-passing backbone to each subgraph and combine the resulting representations through atom-wise aggregation and molecular readout. We evaluated MMGNN on five classification and three regression benchmarks from MoleculeNet using common scaffold splits and five independent runs. MMGNN-2D achieved the highest macro-average AUC-ROC of 0.838 across the classification datasets and the lowest RMSE on ESOL (0.803). MMGNN-3D obtained the highest mean AUC-ROC on BBBP (0.956) and the lowest RMSE on FreeSolv (1.793), indicating complementary strengths of topological and geometric representations. Structural and leave-one-out analyses further illustrate how the subgraph decomposition affects learned representations and atom-type-pair sensitivities. These results support overlapping interaction-specific graph decomposition as a competitive strategy for molecular property prediction.
Trung Nguyen, Duc Duy Nguyen
The Bredesen Center, University of Tennessee, Knoxville, TN 37996, USA · Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
Molecular property prediction often relies on isolated data modalities, where continuous 3D graph neural networks (GNNs) struggle to efficiently capture long-range topological dependencies and exact macroscopic heuristics. In this work, we introduce a parameter-efficient Tri-Branch Modular Fusion Neural Network that synthesizes three orthogonal modalities: 3D spatial geometry (SchNet), discrete topological grammar (SMILES via ChemBERTa), and explicit macroscopic physicochemical descriptors (Deep & Cross Network). By bypassing standard scalar readouts and employing a shared late-fusion architecture, the framework establishes a mathematically rigorous multimodal latent space that effectively resolves the arithmetic and oversmoothing limitations of local message passing. We evaluate the proposed architecture on the QM9 benchmark, targeting the extensive thermodynamic property of atomization energy at 0 K (U0atom). Through systematic combinatorial ablation and latent bottleneck optimization (de=64), the tri-modal framework achieves a validation Mean Absolute Error (MAE) of 0.0207 eV. Operating with fewer than one million parameters, this architecture decisively surpasses the sub-chemical accuracy threshold and yields a substantial 20.6% error reduction over a strictly controlled geometric baseline. Ultimately, our findings demonstrate that integrating orthogonal macroscopic and topological data streams provides a synergistic, O(1) physical shortcut. This multimodal alignment offers a highly efficient alternative to brute-force parameter scaling, establishing a robust surrogate model for high-throughput virtual screening (HTVS) pipelines.
Qiwei Han, Chi Zhou, Ruobing Wang +1
Department of Chemistry, Duke University, Durham, NC 27708, USA · Department of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA
Atomic properties such as partial charges or multipoles encode chemically meaningful information that can inform downstream molecular property prediction, but their evaluation as machine learning targets has been complicated by the absence of a principled out-of-distribution evaluation protocol at the atomic level. In this work, we propose a held-out evaluation protocol that clusters atomic environments by SOAP descriptors and computes metrics accounting only for cluster labels unseen during training. Following this procedure, we use 5×5 cross-validation and Tukey's HSD to run a statistically rigorous comparison of E(3)-equivariant against non-equivariant, rotationally augmented models for predicting electron populations and multipoles of H, C, N, and O atoms. Building on our results, we introduce the Quantum Topological Neural Network (QT-Net), a rotationally augmented, non-equivariant graph neural network. We show that QT-Net can be used to infer properties of atoms in molecules from QM9 outside our training set, and that these inferred properties can yield improvement when used as input features for downstream molecular property prediction. To further validate the framework, molecular dipole moments computed from QT-Net's per-atom outputs recover the ground-truth values reported in QM9. We release all code and data, including a JAX implementation of QT-Net, to support the broader use of learned QTA properties as inductive biases for atomic-scale molecular machine learning.
Pablo Martínez Crespo, Stefano Ribes, Martin Rahm +6
Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg · Department of Chemistry and Chemical Engineering Chalmers University of Technology · Technology Research Intel Corporation +3