TopoResponse

Does persistent homology help an E(3)-equivariant model predict molecular dipole moments and polarizability tensors? A controlled study with matched negative controls, whose answer in this configuration is no.

Course project for the Deep Learning School (DLS). Given a molecule's 3D geometry, predict its dipole moment vector (μ, an ℓ=1 vector) and dipole polarizability tensor (α, ℓ=0 ⊕ ℓ=2) with an E(3)-equivariant network, and study whether persistent-homology features, drawn from topological data analysis (TDA), add predictive value beyond a strong equivariant model — especially under distribution shift. The reported study uses the topology-OOD split; size- and conformation-OOD were scoped but not run.

Why

Equivariant message passing (PaiNN, MACE, …) already predicts molecular response tensors well. Persistent homology offers a global, isometry-invariant descriptor that local message passing can miss. The question is whether that bias measurably helps, and where — the interesting regime is out-of-distribution (OOD) generalization, not the saturated in-distribution benchmark. The prior expectation was a modest accuracy gain and better OOD robustness. It did not materialise (see Status), which is exactly why the matched negative controls were built in from the start: they are what makes a null result informative rather than merely inconclusive.

Data

SQuIRL, the Spectral Quantum Chemistry and Infrared Resonance Library — 133,885 QM9 molecules (ωB97X-D/aug-cc-pVTZ) with the dipole moment vector and full 3×3 polarizability tensor already computed (no re-computation needed). This study uses the 133,883 records present in the processed index; two identifiers (1550 and 65111) are absent from it, hence the small gap against the 133,885 quoted by the dataset paper. Physical sanity checks pass: methane α isotropic; ammonia C₃ᵥ (two equal + one distinct eigenvalue); anisotropic tensors respect point-group symmetry.

Method

• Backbone: E(3)-equivariant network (SchNetPack PaiNN) with dipole (ℓ=1) and polarizability (ℓ=0⊕ℓ=2) heads.
• z_PH descriptor (130-D): Vietoris–Rips persistence of the 3D point cloud, over all atoms and using no element information. Coordinates are centered and divided by the molecular diameter, so every filtration value falls in [0,1] and one fixed 64-point grid is shared by all molecules. The vector concatenates the H₀ and H₁ Betti curves on that grid (64 + 64) with the two persistence entropies.
• TDA conditioning (feature-wise linear modulation): z_PH passes through a small network whose output modulates the backbone's final representation — a per-channel scale and shift on the invariant (scalar) channels, and per-channel invariant multiplicative gates on the equivariant (vector) channels. It is applied after the backbone and never mixes irreducible representations (irreps), so exact E(3) equivariance is preserved. The last layer is zero-initialized, so at initialization the conditioned model reproduces the baseline exactly.
• Splits: random · topology-OOD (train few-ring → test ring-rich) · group-random (molecules grouped by canonical SMILES, whole groups assigned at random).
• Controls run: shuffled z_PH, matched-capacity random features of equal dimension, and an element-augmented 4D persistence variant. TDA and the random/shuffled controls share the same conditioning path and differ only in what it is fed; the baseline has no conditioning path at all. A ring-count baseline, a larger-receptive-field baseline and a separate matched-parameter baseline were scoped but not run.
• Non-equivariant reference models: a simple non-equivariant FCNN on centered, zero-padded coordinates with element one-hots, predicting the dipole components and, in a second run, the nine tensor components scored by Frobenius error; a gradient-boosting model on invariant descriptors (composition, size, ring count, z_PH) predicting the two invariant scalars; and the naive constant predictor (the training mean) as a reference point. These place the equivariant family rather than compete inside it.
• Metrics: dipole vector MAE + angular error; polarizability Frobenius, isotropic/anisotropic split, eigenvalue error; exact equivariance check. The reference models are scored with the same component-wise MAE, and the tabular one with two invariant scalars, |μ| and the isotropic polarizability. Any model without built-in equivariance is additionally evaluated on a rigidly rotated copy of the test set — molecule and reference dipole rotated together, so the task is unchanged and what the rerun measures is orientation sensitivity alone.

Interactive visualization

[NH3+]C1CN2C1C2C([O-])=O, a held-out test molecule. The charge separation between the ammonium and carboxylate ends — blue and red on the density — is exactly what the dipole vector measures. Density: RHF/6-31G at the SQuIRL geometry, isosurface 0.002 e/a₀³; potential from charges fitted to the QM electrostatic potential. Generated by make_density_cubes.py and render_hero.py.*

index.html — a WebGL viewer showing true vs predicted dipole vectors on real molecular geometries, with live angular/magnitude error. Live (GitHub Pages): https://mindvisio.github.io/topo-response/

The seven molecules in the viewer, and the one rendered above, are drawn from the held-out test split of the topology-OOD experiment — ring-rich molecules the model never saw in training — and span 0.6 to 17 D. The errors on display are therefore genuine generalization errors, produced by the seed-0 baseline checkpoint — one of the five seeds the table below aggregates, not the aggregate itself.

Status

The dipole/polarizability study on the topology-OOD split is complete. The tested hypothesis was not supported.

Five seeds per arm (baseline / TDA / matched-capacity random), both properties, paired t-tests on the topology-OOD test set. Raw numbers in results_5seed.csv; recompute with compute_ci.py.

All methods on the same held-out test set, regenerate with make_results_table.py:

model	geometric inductive bias	dipole, compMAE (D)	polarizability, Frobenius (a.u.)
PaiNN baseline	E(3)-equivariant	0.0923 ± 0.0025	2.103 ± 0.169
PaiNN + TDA conditioning	E(3)-equivariant	0.0936 ± 0.0060	2.328 ± 0.137
PaiNN + matched random features	E(3)-equivariant	0.0920 ± 0.0046	2.384 ± 0.277
FCNN on centered padded coordinates	none	0.6593 ± 0.0074	10.194 ± 0.157
naive constant (training mean)	none	1.2920	23.005

Lower is better; ± is the sample standard deviation over the five training seeds. TDA and the random control share one conditioning path and differ only in what it is fed; the baseline has none. The reference models are a comparison, not a matched ablation: they differ from PaiNN in message passing, permutation invariance, architecture and training at once. What rotating the test set does to each is shown separately, on rotation-invariant metrics.

The three equivariant arms sit on top of each other while everything without a geometric inductive bias is several times behind — which is the comparison the paired tests below then quantify. That gap is a reference comparison, not a matched ablation: the reference models differ from PaiNN in message passing, permutation invariance, architecture and training at once, so it cannot be attributed to equivariance alone.

Dipole is component-wise mean absolute error (compMAE) in debye; polarizability is the mean Frobenius error of the 3x3 tensor in atomic units. A positive difference means the first method is worse. Brackets give the paired 95% confidence interval, and the pairing is over the five matched training seeds — not over individual molecules.

paired difference	dipole, compMAE (D)	polarizability, Frobenius (a.u.)
TDA - baseline	+0.0013 [-0.0036, +0.0063], p=0.50	+0.2256 [+0.069, +0.382], p=0.016 (nominal only)
TDA - random	+0.0016 [-0.0056, +0.0088], p=0.57	-0.056 [-0.343, +0.231], p=0.62
random - baseline	-0.0003 [-0.0055, +0.0049], p=0.88	+0.2816 [-0.107, +0.670], p=0.114

No advantage of geometric z_PH conditioning through feature-wise linear modulation (FiLM) over the plain equivariant baseline or over the matched-capacity random control was detected. The one nominally significant effect (polarizability, TDA worse than baseline) does not survive multiple-comparison correction over the six reported tests.

This is a qualitative negative result. A separate bonus experiment (RESIDUAL_PROBE_REPORT.md) freezes the baseline and asks whether z_PH can linearly predict the part of its residual an equivariance-preserving correction may touch; neither that linear probe nor a small nonlinear one detected signal beyond matched random and shuffled controls, consistent with the result above. It does not establish that persistent homology is uninformative or equivalent to noise: no equivalence margin was pre-specified, the confidence intervals remain wide, and the finding does not generalize beyond this descriptor, conditioning scheme, dataset and split. RUN_MANIFEST.md states the caveats in full.

Non-equivariant reference models

The paired tests above compare arms inside the equivariant family. The rows without a geometric inductive bias are what put that family in context.

Those rows are already in the table above. The plain network is 7.1x worse on the dipole and 4.9x worse on the polarizability, closing about half the distance from the naive constant to the equivariant model (53% and 61%). Seed spread is small — 0.6467 to 0.6655 on the dipole — so this is a property of the architecture rather than of one run.

The rotation panel measures the missing symmetry directly. Each test molecule is rigidly rotated together with its reference target, so the task is unchanged, and the comparison uses rotation-invariant metrics: the vector L2 error for the dipole and the Frobenius error for the tensor. That choice matters. Component-wise MAE is an L1 quantity, and an equivariant model rotates its error vector rather than preserving it componentwise, so compMAE shifts by about +1.5% even for PaiNN — a measurement artifact, not a symmetry failure. On the invariant metrics the FCNN loses a further 31% on the dipole and 26% on the polarizability. PaiNN does not move: across all ten runs its error changes by at most 2e-5%, and on half of them by exactly zero.

That last number is worth being precise about, because it is easy to fake. Rotating the stored error vector and observing that its norm is unchanged proves a property of rotation matrices, not of the network. So it is measured the other way round: rotation_forward_check.py feeds each trained checkpoint the rotated geometry and compares the fresh prediction against what equivariance requires, over the whole 66,485-molecule test set. The resulting equivariance defect is 1.4e-06 D (5.4e-07 relative) for the dipole and 2.1e-05 a.u. (1.4e-07 relative) for the tensor — float32 round-off, on all ten runs.

Invariant scalars, where no equivariance is needed at all — a tabular model on composition, size, ring count and z_PH (all five seeds, ± sample sd):

target	PaiNN	gradient boosting	naive constant
dipole magnitude, |μ| (D)	0.1041 ± 0.0025	1.7098	1.2768
isotropic polarizability (a.u.)	0.6028 ± 0.1217	3.7325	6.0741

One entry there deserves a note rather than a footnote: on the shifted test set the tabular model predicts the dipole magnitude worse than the constant. That is not a broken pipeline - on validation, which is drawn from the training regime, the same fitted model beats the constant comfortably (1.0138 against 1.3434). It learns real structure and then fails to extrapolate across the topology shift, and the target distribution moves with it (mean |μ| falls from 3.20 D in training to 2.64 D in test). Polarizability, which tracks molecular size closely, survives the same shift (3.73 against a 6.07 constant). So the split chosen for the main experiment is a demanding one in its own right. What it shows about the dipole is narrower than it looks: the chosen tabular baseline failed to extrapolate across the shift, which is a statement about that baseline, not a proof that geometry is necessary.

Is the descriptor empty?

A null result invites the question of whether z_PH carries any topological information in the first place. It does. Ring count is linearly decodable from the 130-dimensional vector with a held-out R² of 0.62 (mean absolute error 0.57 rings), and the first principal component orders molecules by ring count on its own.

The descriptor is literally the two Betti curves below, sampled on a fixed grid and concatenated with two persistence entropies. H₁ counts independent loops of the atom point cloud, which is not the same object as a chemical ring; empirically its curves still separate by graph-derived ring count on this dataset.

So the negative result is about what the model could use, not about an uninformative input — which is what makes the matched random control the important comparison rather than the baseline alone.

Reproduce

Reproduces from a clean clone, no GPU, no data download:

pip install -r requirements.txt         # or requirements-full.txt for the exact freeze
python compute_ci.py                    # the headline statistics, from the committed CSV
python make_results_table.py --write    # refresh the results table in this README

Needs the excluded artifacts — cache/squirl.db, cache/index.json, cache/zph.npy, cache/split_topology_ood.npz, the trained checkpoints, and the frozen predictions under probe_cache/. Build the caches with the data and feature scripts listed below (set $SQUIRL_H5 to the downloaded archive), the checkpoints per RUN_MANIFEST.md, and probe_cache/ with export_baseline_predictions.py. These were run in the separate CPU environment pinned in requirements-cpu.txt, not the training environment above:

python build_baseline_cache.py
python train_baselines.py all           # dipole + polarizability + tabular + 5-seed scalars
PY=python RUN_MLP=1 bash run_residual_probe.sh          # the bonus probes end to end
for p in dipole polar; do for s in 0 1 2 3 4; do \
    python rotation_forward_check.py --property $p --seed $s; done; done
python make_figures.py                  # all six figures

compute_ci.py reads the committed CSV, so the headline statistics reproduce from a clean clone with no training run and no GPU. Reproducing the checkpoints themselves needs a GPU and the commands in RUN_MANIFEST.md, which also records seeds, hyperparameters, the equivariance check and every caveat attached to the result. Regenerating the viewer assets (density cubes, cover render) additionally needs requirements-assets.txt and a virtual display.

Structure

Data and features

• data_squirl.py — SQuIRL loader (geometry, μ vector, α tensor); build_db.py, build_index.py — database, molecule index, splits
• compute_zph.py — persistent-homology (H₀/H₁) features; compute_zph_elem4d.py — element-augmented 4D variant
• make_grouprandom_split.py — canonical-SMILES group split

Training and evaluation

• train_dipole.py, train_dipole_tda.py, train_p3.py — dipole training (baseline, conditioned, matrix runs)
• train_polar.py, train_p3_polar.py — polarizability training
• eval_ood.py, eval_ood_polar.py — held-out evaluation; e3_test.py — equivariance gate on trained checkpoints
• run_matrix.sh, s4b_run.sh, s4c_run.sh — the seed matrix; compute_ci.py — paired statistics

Bonus: residual probes (see RESIDUAL_PROBE_REPORT.md)

• export_baseline_predictions.py — freeze a baseline and export its predictions
• residual_probe.py (Ridge) · residual_probe_mlp.py (nonlinear) · analyze_residual_probe.py · test_probe_equivariance.py
• run_residual_probe.sh — the whole cycle, with input-integrity checks

Viewer and figures

• index.html — interactive dipole viewer; viewer_infer.py, make_viewer_manifest.py — its predictions and provenance
• make_density_cubes.py — electron density + fitted charges; render_hero.py — the cover image. Visualization only: neither the density nor the electrostatic potential is used as model input
• make_figures.py — the method schematic and the result figures, rebuilt from the committed CSV and z_PH cache
• build_baseline_cache.py, train_baselines.py — the non-equivariant references: an FCNN on centered padded coordinates (scored on rotated copies too) and a tabular model on invariant descriptors
• make_results_table.py — regenerates the results table in this README from the committed CSV and JSON, so the numbers cannot drift; make_figures.py covers all six figures, including the one in the bonus report

Glossary

term	meaning
E(3)	the Euclidean group: rotations, reflections and translations. An equivariant model's output transforms the same way as its input under these
irrep	irreducible representation. Scalars (l=0) and vectors (l=1) transform independently; mixing them would break equivariance
TDA / PH	topological data analysis; persistent homology (PH) is the method within TDA used here
H₀ / H₁	connected components / independent loops, tracked across the filtration
zPH	the 130-dimensional persistence descriptor used throughout: the H₀ and H₁ Betti curves on a fixed 64-point grid, plus their two persistence entropies
OOD	out-of-distribution: the test set is drawn from a different regime than training (here, more rings)
FiLM	feature-wise linear modulation: a learned per-channel scale and shift
PaiNN	Polarizable Atom Interaction Neural Network, the equivariant message-passing backbone
SQuIRL	Spectral Quantum Chemistry and Infrared Resonance Library, the source dataset
MAE / compMAE	mean absolute error; compMAE averages it over the three Cartesian components
Frobenius error	the norm of the difference between predicted and reference 3x3 tensors
R²	fraction of variance explained out of sample; R² ≤ 0 means no better than predicting the mean
CI	confidence interval
MLP	multilayer perceptron
FCNN	fully connected neural network: a plain multilayer perceptron with no geometric structure built in
RHF	restricted Hartree-Fock, the level of theory behind the figure's electron density
ESP	electrostatic potential, used to color that density
D / a.u. / a₀	debye (dipole); atomic units (polarizability, in bohr³); a₀ is the bohr radius
SMILES	a line notation for molecular structure

License and citation

Code and documentation in this repository are released under the MIT License. CITATION.cff gives the citation metadata, which GitHub renders as a Cite this repository button.

That licence covers this repository's own contents. It does not relicence the underlying data: molecular geometries and reference dipoles/polarizabilities come from SQuIRL, and anything derived from them here — the density cubes under dens/, the fitted charges, the results tables — remains subject to the dataset's own terms. Cite SQuIRL alongside this repository if you use them.