Author name cluster

Leon Klein

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

8 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Amortized Sampling with Transferable Normalizing Flows

Charlie Tan
Majdi Hassan
Leon Klein
Saifuddin Syed
Dominique Beaini
Michael Bronstein
Alexander Tong
Kirill Neklyudov

Efficient equilibrium sampling of molecular conformations remains a core challenge in computational chemistry and statistical inference. Classical approaches such as molecular dynamics or Markov chain Monte Carlo inherently lack amortization; the computational cost of sampling must be paid in full for each system of interest. The widespread success of generative models has inspired interest towards overcoming this limitation through learning sampling algorithms. Despite performing competitively with conventional methods when trained on a single system, learned samplers have so far demonstrated limited ability to transfer across systems. We demonstrate that deep learning enables the design of scalable and transferable samplers by introducing Prose, a 285 million parameter all-atom transferable normalizing flow trained on a corpus of peptide molecular dynamics trajectories up to 8 residues in length. Prose draws zero-shot uncorrelated proposal samples for arbitrary peptide systems, achieving the previously intractable transferability across sequence length, whilst retaining the efficient likelihood evaluation of normalizing flows. Through extensive empirical evaluation we demonstrate the efficacy of Prose as a proposal for a variety of sampling algorithms, finding a simple importance sampling-based finetuning procedure to achieve competitive performance to established methods such as sequential Monte Carlo. We open-source the Prose codebase, model weights, and training dataset, to further stimulate research into amortized sampling methods and finetuning objectives.

PDF Details

NeurIPS Conference 2025 Conference Paper

Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models

Michael Plainer
Hao Wu
Leon Klein
Stephan Günnemann
Frank Noe

In recent years, diffusion models trained on equilibrium molecular distributions have proven effective for sampling biomolecules. Beyond direct sampling, the score of such a model can also be used to derive the forces that act on molecular systems. However, while classical diffusion sampling usually recovers the training distribution, the corresponding energy-based interpretation of the learned score is often inconsistent with this distribution, even for low-dimensional toy systems. We trace this inconsistency to inaccuracies of the learned score at very small diffusion timesteps, where the model must capture the correct evolution of the data distribution. In this regime, diffusion models fail to satisfy the Fokker-Planck equation, which governs the evolution of the score. We interpret this deviation as one source of the observed inconsistencies and propose an energy-based diffusion model with a Fokker-Planck-derived regularization term to enforce consistency. We demonstrate our approach by sampling and simulating multiple biomolecular systems, including fast-folding proteins, and by introducing a state-of-the-art transferable Boltzmann emulator for dipeptides that supports simulation and achieves improved consistency and efficient sampling. Our code, model weights, and self-contained JAX and PyTorch notebooks are available at https: //github. com/noegroup/ScoreMD.

PDF Details

NeurIPS Conference 2025 Conference Paper

Path Gradients after Flow Matching

Lorenz Vaitl
Leon Klein

Boltzmann Generators have emerged as a promising machine learning tool for generating samples from equilibrium distributions of molecular systems using Normalizing Flows and importance weighting. Recently, Flow Matching has helped speed up Continuous Normalizing Flows (CNFs), scale them to more complex molecular systems, and minimize the length of the flow integration trajectories. We investigate the benefits of using path gradients to fine-tune CNFs initially trained by Flow Matching, in the setting where a target energy is known. Our experiments show that this hybrid approach yields up to a threefold increase in sampling efficiency for molecular systems, all while using the same model, a similar computational budget and without the need for additional sampling. Furthermore, by measuring the length of the flow trajectories during fine-tuning, we show that path gradients largely preserve the learned structure of the flow.

PDF Details

ICML Conference 2025 Conference Paper

Scalable Equilibrium Sampling with Sequential Boltzmann Generators

Charlie B. Tan
Avishek Joey Bose
Chen Lin 0003
Leon Klein
Michael M. Bronstein
Alexander Tong 0001

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators tackle this problem by pairing normalizing flows with importance sampling to obtain uncorrelated samples under the target distribution. In this paper, we extend the Boltzmann generator framework with two key contributions, denoting our framework Sequential Boltzmann Generators (SBG). The first is a highly efficient Transformer-based normalizing flow operating directly on all-atom Cartesian coordinates. In contrast to the equivariant continuous flows of prior methods, we leverage exactly invertible non-equivariant architectures which are highly efficient during both sample generation and likelihood evaluation. This efficiency unlocks more sophisticated inference strategies beyond standard importance sampling. In particular, we perform inference-time scaling of flow samples using a continuous-time variant of sequential Monte Carlo, in which flow samples are transported towards the target distribution with annealed Langevin dynamics. SBG achieves state-of-the-art performance w. r. t. all metrics on peptide systems, demonstrating the first equilibrium sampling in Cartesian coordinates of tri-, tetra- and hexa-peptides that were thus far intractable for prior Boltzmann generators.

Details

NeurIPS Conference 2024 Conference Paper

Transferable Boltzmann Generators

Leon Klein
Frank Noé

The generation of equilibrium samples of molecular systems has been a long-standing problem in statistical physics. Boltzmann Generators are a generative machine learning method that addresses this issue by learning a transformation via a normalizing flow from a simple prior distribution to the target Boltzmann distribution of interest. Recently, flow matching has been employed to train Boltzmann Generators for small molecular systems in Cartesian coordinates. We extend this work and propose a first framework for Boltzmann Generators that are transferable across chemical space, such that they predict zero-shot Boltzmann distributions for test molecules without being retraining for these systems. These transferable Boltzmann Generators allow approximate sampling from the target distribution of unseen systems, as well as efficient reweighting to the target Boltzmann distribution. The transferability of the proposed framework is evaluated on dipeptides, where we show that it generalizes efficiently to unseen systems. Furthermore, we demonstrate that our proposed architecture enhances the efficiency of Boltzmann Generators trained on single molecular systems.

PDF Details DOI

NeurIPS Conference 2023 Conference Paper

Equivariant flow matching

Leon Klein
Andreas Krämer
Frank Noe

Normalizing flows are a class of deep generative models that are especially interesting for modeling probability distributions in physics, where the exact likelihood of flows allows reweighting to known target energy functions and computing unbiased observables. For instance, Boltzmann generators tackle the long-standing sampling problem in statistical physics by training flows to produce equilibrium samples of many-body systems such as small molecules and proteins. To build effective models for such systems, it is crucial to incorporate the symmetries of the target energy into the model, which can be achieved by equivariant continuous normalizing flows (CNFs). However, CNFs can be computationally expensive to train and generate samples from, which has hampered their scalability and practical application. In this paper, we introduce equivariant flow matching, a new training objective for equivariant CNFs that is based on the recently proposed optimal transport flow matching. Equivariant flow matching exploits the physical symmetries of the target energy for efficient, simulation-free training of equivariant CNFs. We demonstrate the effectiveness of flow matching on rotation and permutation invariant many-particle systems and a small molecule, alanine dipeptide, where for the first time we obtain a Boltzmann generator with significant sampling efficiency without relying on tailored internal coordinate featurization. Our results show that the equivariant flow matching objective yields flows with shorter integration paths, improved sampling efficiency, and higher scalability compared to existing methods.

PDF Details

NeurIPS Conference 2023 Conference Paper

Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics

Leon Klein
Andrew Foong
Tor Fjelde
Bruno Mlodozeniec
Marc Brockschmidt
Sebastian Nowozin
Frank Noe
Ryota Tomioka

*Molecular dynamics* (MD) simulation is a widely used technique to simulate molecular systems, most commonly at the all-atom resolution where equations of motion are integrated with timesteps on the order of femtoseconds ($1\textrm{fs}=10^{-15}\textrm{s}$). MD is often used to compute equilibrium properties, which requires sampling from an equilibrium distribution such as the Boltzmann distribution. However, many important processes, such as binding and folding, occur over timescales of milliseconds or beyond, and cannot be efficiently sampled with conventional MD. Furthermore, new MD simulations need to be performed for each molecular system studied. We present *Timewarp*, an enhanced sampling method which uses a normalising flow as a proposal distribution in a Markov chain Monte Carlo method targeting the Boltzmann distribution. The flow is trained offline on MD trajectories and learns to make large steps in time, simulating the molecular dynamics of $10^{5} - 10^{6} \textrm{fs}$. Crucially, Timewarp is *transferable* between molecular systems: once trained, we show that it generalises to unseen small peptides (2-4 amino acids) at all-atom resolution, exploring their metastable states and providing wall-clock acceleration of sampling compared to standard MD. Our method constitutes an important step towards general, transferable algorithms for accelerating MD.

PDF Details

ICML Conference 2020 Conference Paper

Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities

Jonas Köhler 0001
Leon Klein
Frank Noé

Normalizing flows are exact-likelihood generative neural networks which approximately transform samples from a simple prior distribution to samples of the probability distribution of interest. Recent work showed that such generative models can be utilized in statistical mechanics to sample equilibrium states of many-body systems in physics and chemistry. To scale and generalize these results, it is essential that the natural symmetries in the probability density – in physics defined by the invariances of the target potential – are built into the flow. We provide a theoretical sufficient criterion showing that the distribution generated by equivariant normalizing flows is invariant with respect to these symmetries by design. Furthermore, we propose building blocks for flows which preserve symmetries which are usually found in physical/chemical many-body particle systems. Using benchmark systems motivated from molecular physics, we demonstrate that those symmetry preserving flows can provide better generalization capabilities and sampling efficiency.

Details