Bálint Máté

Bálint Máté, Samuel Klein, Tobias Golling et al.

The two key characteristics of a normalizing flow is that it is invertible (in particular, dimension preserving) and that it monitors the amount by which it changes the likelihood of data points as samples are propagated along the network. Recently, multiple generalizations of normalizing flows have been introduced that relax these two conditions. On the other hand, neural networks only perform a forward pass on the input, there is neither a notion of an inverse of a neural network nor is there one of its likelihood contribution. In this paper we argue that certain neural network architectures can be enriched with a stochastic inverse pass and that their likelihood contribution can be monitored in a way that they fall under the generalized notion of a normalizing flow mentioned above. We term this enrichment flowification. We prove that neural networks only containing linear layers, convolutional layers and invertible activations such as LeakyReLU can be flowified and evaluate them in the generative setting on image datasets.

Bálint Máté, François Fleuret

We introduce a training objective for continuous normalizing flows that can be used in the absence of samples but in the presence of an energy function. Our method relies on either a prescribed or a learnt interpolation $f_t$ of energy functions between the target energy $f_1$ and the energy function of a generalized Gaussian $f_0(x) = ||x/σ||_p^p$. The interpolation of energy functions induces an interpolation of Boltzmann densities $p_t \propto e^{-f_t}$ and we aim to find a time-dependent vector field $V_t$ that transports samples along the family $p_t$ of densities. The condition of transporting samples along the family $p_t$ is equivalent to satisfying the continuity equation with $V_t$ and $p_t = Z_t^{-1}e^{-f_t}$. Consequently, we optimize $V_t$ and $f_t$ to satisfy this partial differential equation. We experimentally compare the proposed training objective to the reverse KL-divergence on Gaussian mixtures and on the Boltzmann density of a quantum mechanical particle in a double-well potential.

Bálint Máté, Bertrand Le Saux, Maxwell Henderson

This paper explores the advantages of optimizing quantum circuits on $N$ wires as operators in the unitary group $U(2^N)$. We run gradient-based optimization in the Lie algebra $\mathfrak u(2^N)$ and use the exponential map to parametrize unitary matrices. We argue that $U(2^N)$ is not only more general than the search space induced by an ansatz, but in ways easier to work with on classical computers. The resulting approach is quick, ansatz-free and provides an upper bound on performance over all ansätze on $N$ wires.

Daniele Paliotta, Mathieu Alain, Bálint Máté et al.

We present the Graph Forward-Forward (GFF) algorithm, an extension of the Forward-Forward procedure to graphs, able to handle features distributed over a graph's nodes. This allows training graph neural networks with forward passes only, without backpropagation. Our method is agnostic to the message-passing scheme, and provides a more biologically plausible learning scheme than backpropagation, while also carrying computational advantages. With GFF, graph neural networks are trained greedily layer by layer, using both positive and negative samples. We run experiments on 11 standard graph property prediction tasks, showing how GFF provides an effective alternative to backpropagation for training graph neural networks. This shows in particular that this procedure is remarkably efficient in spite of combining the per-layer training with the locality of the processing in a GNN.

Bálint Máté, François Fleuret

Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. This work studies the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $Ψ_{t_1}^{t_0}$ so that the transformed samples follow $p_{t_0}$. We derive an equation relating $Ψ$ and the corresponding family of unnormalized log-likelihoods $S_t$. The utility of this idea is demonstrated on the $φ^4$ lattice field theory by extending its defining action $S_0$ to a family of actions $S_t$ and finding a $τ$ such that normalizing flows perform better at learning the Boltzmann distribution $p_τ$ than at learning $p_0$.

Atul Kumar Sinha, Daniele Paliotta, Bálint Máté et al.

Deep learning methods have gained popularity in high energy physics for fast modeling of particle showers in detectors. Detailed simulation frameworks such as the gold standard Geant4 are computationally intensive, and current deep generative architectures work on discretized, lower resolution versions of the detailed simulation. The development of models that work at higher spatial resolutions is currently hindered by the complexity of the full simulation data, and by the lack of simpler, more interpretable benchmarks. Our contribution is SUPA, the SUrrogate PArticle propagation simulator, an algorithm and software package for generating data by simulating simplified particle propagation, scattering and shower development in matter. The generation is extremely fast and easy to use compared to Geant4, but still exhibits the key characteristics and challenges of the detailed simulation. We support this claim experimentally by showing that performance of generative models on data from our simulator reflects the performance on a dataset generated with Geant4. The proposed simulator generates thousands of particle showers per second on a desktop machine, a speed up of up to 6 orders of magnitudes over Geant4, and stores detailed geometric information about the shower propagation. SUPA provides much greater flexibility for setting initial conditions and defining multiple benchmarks for the development of models. Moreover, interpreting particle showers as point clouds creates a connection to geometric machine learning and provides challenging and fundamentally new datasets for the field. The code for SUPA is available at https://github.com/itsdaniele/SUPA.

Giulia Luise, Chin-Wei Huang, Thijs Vogels et al.

Density Functional Theory (DFT) is the most widely used electronic structure method for predicting the properties of molecules and materials. Although DFT is, in principle, an exact reformulation of the Schrödinger equation, practical applications rely on approximations to the unknown exchange-correlation (XC) functional. Most existing XC functionals are constructed using a limited set of increasingly complex, hand-crafted features that improve accuracy at the expense of computational efficiency. Yet, no current approximation achieves the accuracy and generality for predictive modeling of laboratory experiments at chemical accuracy -- typically defined as errors below 1 kcal/mol. In this work, we present Skala, a modern deep learning-based XC functional that bypasses expensive hand-designed features by learning representations directly from data. Skala achieves chemical accuracy for atomization energies of small molecules while retaining the computational efficiency typical of semi-local DFT. This performance is enabled by training on an unprecedented volume of high-accuracy reference data generated using computationally intensive wavefunction-based methods. Notably, Skala systematically improves with additional training data covering diverse chemistry. By incorporating a modest amount of additional high-accuracy data tailored to chemistry beyond atomization energies, Skala achieves accuracy competitive with the best-performing hybrid functionals across general main group chemistry, at the cost of semi-local DFT. As the training dataset continues to expand, Skala is poised to further enhance the predictive power of first-principles simulations.

Bálint Máté, François Fleuret, Tristan Bereau

We present a method for computing free-energy differences using thermodynamic integration with a neural network potential that interpolates between two target Hamiltonians. The interpolation is defined at the sample distribution level, and the neural network potential is optimized to match the corresponding equilibrium potential at every intermediate time-step. Once the interpolating potentials and samples are well-aligned, the free-energy difference can be estimated using (neural) thermodynamic integration. To target molecular systems, we simultaneously couple Lennard-Jones and electrostatic interactions and model the rigid-body rotation of molecules. We report accurate results for several benchmark systems: a Lennard-Jones particle in a Lennard-Jones fluid, as well as the insertion of both water and methane solutes in a water solvent at atomistic resolution using a simple three-body neural-network potential.

Juno Nam, Bálint Máté, Artur P. Toshev et al.

Diffusion-based samplers learn to sample complex, high-dimensional distributions using energies or log densities alone, without training data. Yet, they remain impractical for molecular sampling because they are often slower than molecular dynamics and miss thermodynamically relevant modes. Inspired by enhanced sampling, we encourage exploration by introducing a sequential bias along bespoke, information-rich, low-dimensional projections of atomic coordinates known as collective variables (CVs). We introduce a repulsive potential centered on the CVs from recent samples, which pushes future samples towards novel CV regions and effectively increases the temperature in the projected space. Our resulting method improves efficiency, mode discovery, enables the estimation of free energy differences, and retains independent sampling from the approximate Boltzmann distribution via reweighting by the bias. On standard peptide conformational sampling benchmarks, the method recovers diverse conformational states and accurate free energy profiles. We are the first to demonstrate reactive sampling using a diffusion-based sampler, capturing bond breaking and formation with universal interatomic potentials at near-first-principles accuracy. The approach resolves reactive energy landscapes at a fraction of the wall-clock time of standard sampling methods, advancing diffusion-based sampling towards practical use in molecular sciences.

Bálint Máté, François Fleuret

We consider the problem of sampling lattice field configurations on a lattice from the Boltzmann distribution corresponding to some action. Since such densities arise as approximationw of an underlying functional density, we frame the task as an instance of operator learning. We propose to approximate a time-dependent neural operator whose time integral provides a mapping between the functional distributions of the free and target theories. Once a particular lattice is chosen, the neural operator can be discretized to a finite-dimensional, time-dependent vector field which in turn induces a continuous normalizing flow between finite dimensional distributions over the chosen lattice. This flow can then be trained to be a diffeormorphism between the discretized free and target theories on the chosen lattice, and, by construction, can be evaluated on different discretizations of spacetime. We experimentally validate the proposal on the 2-dimensional $φ^4$-theory to explore to what extent such operator-based flow architectures generalize to lattice sizes they were not trained on, and show that pretraining on smaller lattices can lead to a speedup over training directly on the target lattice size.

Bálint Máté, François Fleuret, Tristan Bereau

Thermodynamic integration (TI) offers a rigorous method for estimating free-energy differences by integrating over a sequence of interpolating conformational ensembles. However, TI calculations are computationally expensive and typically limited to coupling a small number of degrees of freedom due to the need to sample numerous intermediate ensembles with sufficient conformational-space overlap. In this work, we propose to perform TI along an alchemical pathway represented by a trainable neural network, which we term Neural TI. Critically, we parametrize a time-dependent Hamiltonian interpolating between the interacting and non-interacting systems, and optimize its gradient using a score matching objective. The ability of the resulting energy-based diffusion model to sample all intermediate ensembles allows us to perform TI from a single reference calculation. We apply our method to Lennard-Jones fluids, where we report accurate calculations of the excess chemical potential, demonstrating that Neural TI reproduces the underlying changes in free energy without the need for simulations at interpolating Hamiltonians.

Bálint Máté, François Fleuret

We introduce primed-PCA (pPCA), a two-step algorithm for speeding up the approximation of principal components. This algorithm first runs any approximate-PCA method to get an initial estimate of the principal components (priming), and then applies an exact PCA in the subspace they span. Since this subspace is of small dimension in any practical use, the second step is extremely cheap computationally. Nonetheless, it improves accuracy significantly for a given computational budget across datasets. In this setup, the purpose of the priming is to narrow down the search space, and prepare the data for the second step, an exact calculation. We show formally that pPCA improves upon the priming algorithm under very mild conditions, and we provide experimental validation on both synthetic and real large-scale datasets showing that it systematically translates to improved performance. In our experiments we prime pPCA by several approximate algorithms and report an average speedup by a factor of 7.2 over Oja's rule, and a factor of 10.5 over EigenGame.