The gifs didn't post properly 😅

Here is one showing the electron cloud in two stages: (1) the learning of electron density during training and (2) the predicted ground-state across conformations 😎

(9/9)⚡ Runtime efficiency
Self-refining training reduces total runtime up to 4 times compared to the baseline
and up to 2 times compared to the fully-supervised approach!!!
Less need for large pre-generated datasets — training and sampling happen in parallel.

(8/n) 🧪 Robust generalization
We simulate molecular dynamics using each model’s energy predictions and evaluate accuracy along the trajectory.
Models trained with self-refinement stay accurate even far from the training distribution — while baselines quickly degrade.

(7/n) 📊 Performance under data scarcity
Our method achieves low energy error with as few as 25 conformations.
With 10× less data, it matches or outperforms fully supervised baselines.
This is especially important in settings where labeled data is expensive or unavailable.

(6/n) This minimization leads to Self-Refining Training:
🔁 Use the current model to sample conformations via MCMC
📉 Use those conformations to minimize energy and update the model

Everything runs asynchronously, without need for labeled data and minimal number of conformations from a dataset!

(5/n) To get around this, we introduce a variational upper bound on the KL between any sampling distribution q(R) and the target Boltzmann distribution.

Jointly minimizing this bound wrt θ and q yields
✅ A model that predicts the ground-state solutions
✅ Samples that match the ground true density

(4/n) With an amortized DFT model f_θ(R), we define the density of molecular conformations as the
Boltzmann distribution

This isn't a typical ML setup because
❌ No samples from the density - can’t train a generative model
❌ No density - can’t sample via Monte Carlo!

(3/n) DFT offers a scalable solution to the Schrödinger equation but must be solved independently for each geometry by minimizing energy wrt coefficients C for a fixed basis.

This presents a bottleneck for MD/sampling.

We want to amortize this - train a model that generalizes across geometries R.

(2/n) This work is the result of an amazing collaboration with @fntwin.bsky.social Hatem Helal @dom-beaini.bsky.social @k-neklyudov.bsky.social