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Mathematical Invariant-Enabled Topological Neural Networks for Molecular and Materials Property Prediction

Yiming Ren, Xiang Liu, Mustafa Hajij, Pietro Liò, Guo-Wei Wei

arXiv:2610.07712v1q-bio.BMcs.LG

TL;DR

Existing molecular and materials approaches may capture only selected aspects of complex 3D structure, leaving the value of combining mathematical views and neural architectures an open question. MITNN integrates five invariant families with four neural architectures and evaluates their pairings across molecular and materials benchmarks. The study reports consistent gains over existing methods, while finding that performance depends on which components are paired and integrated.

  • Problem

    Existing approaches may not fully capture complex 3D organization, stereochemistry, and higher-order relationships, and the value of combining mathematical views with different architectures remains unclear.

    Source: §1 · pp. 2–3

  • Method

    MITNN combines five mathematical invariant families with four neural architectures in a framework for molecular and materials property prediction.

    Source: §1 · p. 3

  • Results

    Across four prediction tasks, MITNN consistently outperformed existing methods, while selected invariant–architecture subsets achieved the highest performance.

    Source: §1 · p. 3, §3 · p. 11

  • Takeaways & Limitations

    Predictive gains depend on how mathematical invariants and neural architectures are paired and integrated, rather than simply on broader aggregation.

    Source: §3 · p. 11

    The conclusions are conditioned on the invariant families, architectures, and benchmark applications considered and should not be interpreted as universal across prediction problems.

    Source: §3 · p. 11

Abstract

from arXiv · show

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.

1 Introduction

Molecular and materials models face a representation challenge: existing approaches may miss aspects of complex 3D structure or trade structural fidelity against invariance and computational complexity. MITNNs address this by systematically combining complementary mathematical invariants and neural architectures, then evaluating their performance across four prediction tasks.

1 Introduction

  • Existing representations may not capture 3D organization, stereochemistry, and higher-order relationships across the local, mesoscopic, and global scales.Fixed-length descriptors may omit spatial relationships, while geometric models must account for permutation, translation, and rotation symmetries; pairwise message passing may miss angular and many-body organization.Source: §1 · p. 2
  • The study asks whether combining mathematical views improves learning and how each view’s utility depends on the architecture processing it.Prior work generally focused on individual representations or selected invariant–architecture combinations, leaving their complementarity and interactions unclear.Source: §1 · p. 3
  • MITNNs pair five invariant families with four neural architectures and benchmark the resulting models across four molecular and materials prediction tasks.The tasks are protein–ligand binding affinity, MOF gas uptake, mutation-induced protein solubility, and quantitative toxicity prediction.Source: §1 · p. 3
  • MITNNs combine persistent homology, persistent Laplacian, commutative algebra, element interactive curvature, and Forman persistent Ricci curvature with ANNs, CNNs, SNNs, and CTNNs.The invariant families represent topological, spectral, algebraic–combinatorial, geometric, and discrete-curvature information.Source: §1 · pp. 2–3
  • Across the four tasks, MITNNs consistently outperform existing state-of-the-art methods.The framework evaluates invariant and architecture roles and interactions rather than assuming one component is universally optimal.Source: §1 · p. 3

2 Results

MITNN maps molecular and materials structures into mathematical domains, constructs multiscale invariants, processes them with neural architectures, and aggregates base-model predictions. Its framework allows different invariant–architecture pairings and consensus strategies to be evaluated.

2.1 Overview of the MITNN Framework

  • MITNN maps 3D structures to mathematical domains, constructs invariants, processes their features with neural architectures, and aggregates base-model predictions.The framework adapts domains and invariant constructions to structurally distinct applications, including protein–ligand complexes, MOFs, proteins, and small molecules.Source: §2.1 · p. 3
  • The framework uses five invariant families: CA, PH, PL, EIC, and FPRC, capturing algebraic–combinatorial, topological, spectral, geometric, and discrete-curvature information.PH tracks topological features across filtration scales; PL adds spectral characteristics, while EIC and FPRC describe continuous and discrete curvature, respectively.Source: §2.1 · pp. 3, 5
  • Four architectures process the invariant features: SNNs and CTNNs organize higher-order information, while ANNs and CNNs provide conventional neural processing.CTNNs use copresheaf structure maps and low-rank copresheaf transformer self-attention; CNNs process ordered multiscale feature arrays.Source: §2.1 · p. 5
  • Pairing five invariant families with four architectures yields up to 20 base models, which MITNN aggregates as MITNNall or MITNNbest.MITNNall aggregates all available base models; MITNNbest uses the best-performing subset identified through exhaustive ensemble analysis.Source: Figure 1 · p. 4, §2.1 · p. 5

2.2 MITNN for Protein–Ligand Binding Affinity Prediction

On CASF-2016, MITNNbest outperformed the evaluated scoring functions and topology-based learning methods, achieving the highest PCC and lowest reported SD. The benchmark uses refined PDBbind-v2016 complexes for training and CASF-2016 core complexes for evaluation.

2.2 MITNN for Protein–Ligand Binding Affinity Prediction

  • The study evaluated binding-affinity prediction on CASF-2016, a benchmark used to assess computational scoring performance.Binding affinity quantifies interaction strength between a ligand and its macromolecular target and is a central endpoint in structure-based drug discovery.Source: §2.2 · p. 5
  • The evaluation trained on PDBbind-v2016 refined complexes and tested on CASF-2016 core complexes.Each base model was trained in ten independent runs; consensus predictions used the median across included base models.Source: §2.2 · p. 5
  • PCC rose from 0.852 to 0.865 and SD fell from 1.612 to 1.483 kcal/mol versus TopoFormer-Seq, a 1.5% relative PCC improvement and 8.0% SD reduction.MITNNbest achieved the highest PCC and lowest reported SD among evaluated methods; MITNNall achieved a PCC of 0.857 and an SD of 1.522 kcal/mol.Source: §2.2 · p. 5

2.3 MITNN for MOF Gas-Uptake Prediction

On MOF gas-uptake prediction, both MITNN consensus models outperformed the evaluated descriptor-based, topology-based, and transformer-based approaches. MITNNbest improved performance over the strongest non-MITNN baselines for both O2 and N2 uptake.

2.3 MITNN for MOF Gas-Uptake Prediction

  • MOF gas adsorption depends on global structural features, including pore geometry and framework connectivity, and on local chemical environments.The study grouped constituent elements into eight chemical categories plus an all-atom category to account for elemental heterogeneity.Source: §2.3 · pp. 5, 7
  • The evaluation predicted O2 and N2 uptake in mol/kg using ten random train–validation–test partitions with an 8:1:1 ratio.Reported R2, MAE, and RMSE metrics were averaged across the ten partitions.Source: §2.3 · p. 7
  • MITNNbest improved R2 by 7.9% for O2 uptake and 7.0% for N2 uptake versus the strongest non-MITNN baselines.It also reduced MAE by 23.9% and 15.1%, and RMSE by 23.3% and 13.8%, for O2 and N2, respectively.Source: Figure 2 · p. 6, §2.3 · p. 7
  • For O2 uptake, R2 increased from 0.909 with MITNNall to 0.917 with MITNNbest.For N2 uptake, the corresponding R2 values were 0.838 and 0.845; MITNNbest also yielded lower MAE and RMSE for both gases.Source: §2.3 · p. 7

2.4 MITNN for Mutation-Induced Protein Solubility Prediction

On the 662-variant PON-Sol2 blind test, MITNNbest outperformed the published predictor on both reported metrics.

2.4 MITNN for Mutation-Induced Protein Solubility Prediction

  • The independent position-stratified blind test comprised 662 variants: 338 decreased-solubility, 87 neutral, and 237 increased-solubility mutations.The task is three-class classification.Source: §2.4 · p. 7
  • Each invariant–architecture base model was trained in ten random-seed runs, with predictions averaged before class assignment.Performance was evaluated with CPR and GC2.Source: §2.4 · p. 7
  • A prior result achieved CPR 0.669 and GC2 0.149; MITNNbest improved them to 0.699 and 0.184, outperforming published PON-Sol2, which reported CPR 0.671 and GC2 0.181.Source: §2.4 · p. 7

2.5 MITNN for Quantitative Toxicity Prediction

For oral rat LD50 prediction, MITNNbest achieved the strongest reported performance among evaluated methods, while the accompanying analyses examine invariant and architecture complementarity.

2.5 MITNN for Quantitative Toxicity Prediction

  • MITNNbest achieved PCC2 0.684 and RMSE 0.542, the highest PCC2 and lowest RMSE among evaluated toxicity methods.MITNNall achieved PCC2 0.675 and RMSE 0.550.Source: §2.5 · p. 7
  • Base models used ten independent random-seed runs; MITNNall and MITNNbest predictions were formed by successive medians across runs and included models.The metrics were PCC2 and RMSE.Source: §2.5 · p. 7
  • The LD50 invariant-subset analysis reports PCC2 across four architectures, while the ensemble landscape compares candidate subset sizes and marks MITNNbest and MITNNall.Figure 3 also describes subset-size performance for LD50; Figure 4 includes LD50 among its ensemble applications.Source: Figure 3 · p. 8, Figure 4 · p. 9

2.6 Mathematical-Invariant Complementarity within Fixed Neural Architectures

Invariant combinations often improve performance, but their value depends on composition and on the application and architecture.

2.6 Mathematical-Invariant Complementarity within Fixed Neural Architectures

  • The analysis tested all 31 nonempty subsets of five invariant families for CASF-2016, PON-Sol2, and LD50, and all 15 subsets of four families for MOF uptake.MOF complementarity results here focus on O2 uptake; N2 results are reported separately.Source: §2.6 · p. 10
  • Combining invariant families generally improved performance, with the largest average gain usually occurring from one to two invariants and smaller gains thereafter.For CASF-2016 CNN, PH+PL+CA+EIC reached PCC 0.853, slightly above 0.852 with all five families.Source: §2.6 · p. 10
  • Invariant preferences varied by application: PL recurred in strong CASF-2016 subsets, while CA and FPRC were frequent for MOF O2 uptake.PH or PL added gains for MOF O2 uptake depending on architecture; PON-Sol2 and LD50 combinations varied more across architectures.Source: §2.6 · p. 10
  • Across applications, CNN had the highest central tendency for CASF-2016, CTNN for MOF O2 uptake and LD50, and PON-Sol2 distributions were more similar across architectures.Source: §2.6 · p. 10

2.7 Architectural Complementarity for Fixed Mathematical Invariants

Combining architectures usually improves average performance, but gains are nonmonotonic and depend on which architectures and invariant are paired.

2.7 Architectural Complementarity for Fixed Mathematical Invariants

  • Across most invariant–application settings, architecture combinations improved average performance over single architectures, but gains were not necessarily monotonic.All 15 nonempty subsets of ANN, CNN, SNN, and CTNN were evaluated for each invariant.Source: §2.7 · p. 10
  • PL stayed strong across CASF-2016 architecture subsets; CA led MOF O2 uptake, while FPRC gained more from combining architectures.Source: §2.7 · p. 10
  • For PON-Sol2, architecture composition mattered more than subset size; for LD50, FPRC was strong with CTNN alone while EIC benefited more from combinations.Source: §2.7 · p. 10

2.8 Selective Integration of Invariant–Architecture Base Models

Ensemble performance generally improves and becomes less variable as more invariant–architecture models are included, but the best performance comes from selected subsets. Which pairings are favored varies across applications.

2.8 Selective Integration of Invariant–Architecture Base Models

  • Following invariant–architecture complementarity analyses, the study examined how ensemble size and composition affect performance using application-specific aggregation procedures.Candidate subsets were evaluated with the aggregation procedures defined for each application.Source: §2.8 · pp. 10–11
  • 9, 7, 6, and 7 base models produced the best ensembles for CASF-2016, MOF O2 uptake, PON-Sol2, and LD50, respectively, before all available models were included.Median performance rose rapidly for smaller ensembles and then approached a plateau, while variability narrowed as ensemble size increased.Source: §2.8 · p. 11
  • High-performing ensembles favored particular invariant–architecture pairings, with PL-based models frequent for CASF-2016 and CA- and FPRC-based models more prominent for MOF O2 uptake.Inclusion frequencies among the top 100 ensembles were nonuniform and varied across applications.Source: §2.8 · p. 11

3 Discussion

MITNN combines multiple mathematical representations with several neural architectures for molecular and materials prediction. The discussion reports strong performance across the evaluated applications, while emphasizing that results depend on pairings and remain bounded by the systems and benchmarks considered.

3 Discussion

  • MITNN combines five invariant families with topological, artificial, and convolutional neural architectures in a shared library of invariant–architecture base models.The framework is evaluated for protein–ligand binding affinity, MOF gas uptake, mutation-induced protein solubility, and quantitative toxicity, with consistently strong performance relative to evaluated baselines.Source: §3 · p. 11
  • Predictive performance depends jointly on the invariant and architecture, and the benefit of combining them varies with the specific pairing and prediction task.For CASF-2016, frequent PL-based models among high-performing combinations may reflect spectral information alongside multiscale topological connectivity.Source: §3 · p. 11
  • Broader aggregation improves typical performance and reduces variability, but the highest performance comes from selected invariant–architecture subsets.High-performing ensembles preferentially represent certain pairings, so gains depend on how mathematical and architectural diversity is integrated.Source: §3 · p. 11
  • The conclusions are conditioned on the invariant families, neural architectures, and benchmark applications considered, and should not be treated as universal across prediction problems.Source: §3 · p. 11
  • Possible extensions include additional mathematical constructions and neural architectures; a growing invariant–architecture library will also require scalable base-model screening and selection.The discussion also identifies potential extension to other complex scientific data involving complementary mathematical representations and high-order neural processing.Source: §3 · pp. 11–12

4 Methods

The methods define five scale-dependent invariant families, built from filtered simplicial complexes or element-specific density fields, with invariance depending on the construction.

4.1 Mathematical Invariants

  • The five invariant families capture topology, Laplacian spectra, algebraic–combinatorial structure, element-specific geometry, and discrete curvature across scales.Their values vary with scale while remaining invariant under admissible transformations at each fixed scale.Source: §4.1 · p. 12

4.2 Simplicial Complexes and Filtrations

  • Nested simplicial complexes across filtration scales provide the shared basis for topological, spectral, algebraic–combinatorial, and discrete-curvature invariants.The schematic Vietoris–Rips example shows complexes changing as the filtration parameter increases.Source: §4.2 · p. 13, Figure 5 · p. 13

4.3 Persistent Homology (PH)

  • Persistent homology summarizes topological features through birth–death intervals, with translations and rotations preserving distance-based barcodes but uniform scaling not treated as invariant.The illustrated H0 and H1 intervals track component merging and the formation and filling of loops.Source: §4.3 · p. 14

4.4 Persistent Laplacian (PL)

  • Persistent Laplacian spectra retain Betti-number information in zero eigenvalues and add complementary information in positive eigenvalues.For the illustrated filtration, positive eigenvalues emerge as edges appear and coincide at the complete clique complex.Source: §4.4 · pp. 14–15

4.5 Commutative Algebra (CA)

  • Commutative-algebra features track facet persistence and f-vectors, which capture complementary aspects of filtered complexes but are not complete invariants.Facet bars follow maximal simplices as they appear and become faces of higher-dimensional simplices.Source: §4.5 · pp. 15–16

4.6 Element Interactive Curvature (EIC)

  • Element interactive curvature derives local geometric quantities from differentiable, element-specific interaction-density fields across interaction scales.The construction is invariant to atom reindexing, translations, and rotations under a consistent normal convention; its distance constraint excludes covalent interactions.Source: §4.6 · pp. 16–17

4.7 Forman Persistent Ricci Curvature (FPRC)

  • Forman persistent Ricci curvature tracks local face–coface and parallel-simplex relations across a filtration, and the example’s mean-curvature curves converge at the complete clique complex.Isolated vertices are assigned zero curvature; the first edge and triangle have curvatures 2 and 3, respectively.Source: §4.7 · pp. 17–18

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