Linear systems arise in generating samples and in calculating observables in lattice quantum chromodynamics~(QCD). Solving the Hermitian positive definite systems, which are sparse but ill-conditioned, involves using iterative methods, such as Conjugate Gradient (CG), which are time-consuming and computationally expensive. Preconditioners can effectively accelerate this process, with the state-of-the-art being multigrid preconditioners. However, constructing useful preconditioners can be challenging, adding additional computational overhead, especially in large linear systems. We propose a framework, leveraging operator learning techniques, to construct linear maps as effective preconditioners. The method in this work does not rely on explicit matrices from either the original linear systems or the produced preconditioners, allowing efficient model training and application in the CG solver. In the context of the Schwinger model U(1) gauge theory in 1+1 spacetime dimensions with two degenerate-mass fermions), this preconditioning scheme effectively decreases the condition number of the linear systems and approximately halves the number of iterations required for convergence in relevant parameter ranges. We further demonstrate the framework learns a general mapping dependent on the lattice structure which leads to zero-shot learning ability for the Dirac operators constructed from gauge field configurations of different sizes.

We develop diffusion models for simulating lattice gauge theories, where stochastic quantization is explicitly incorporated as a physical condition for sampling. We demonstrate the applicability of this novel sampler to U(1) gauge theory in two spacetime dimensions and find that a model trained at a small inverse coupling constant can be extrapolated to larger inverse coupling regions without encountering the topological freezing problem. Additionally, the trained model can be employed to sample configurations on different lattice sizes without requiring further training. The exactness of the generated samples is ensured by incorporating Metropolis-adjusted Langevin dynamics into the generation process. Furthermore, we demonstrate that this approach enables more efficient sampling of topological quantities compared to traditional algorithms such as Hybrid Monte Carlo and Langevin simulations.

We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers. Inverse renormalization group methods were originally established as generative numerical techniques by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations [1].

Deep generative models complement Markov-chain-Monte-Carlo methods for efficiently sampling from high-dimensional distributions. Among these methods, explicit generators, such as Normalising Flows (NFs), in combination with the Metropolis Hastings algorithm have been extensively applied to get unbiased samples from target distributions. We systematically study central problems in conditional NFs, such as high variance, mode collapse and data efficiency. We propose adversarial training for NFs to ameliorate these problems. Experiments are conducted with low-dimensional synthetic datasets and XY spin models in two spatial dimensions.

We consider the problem of sampling discrete field configurations $\phi$ from the Boltzmann distribution $[d\phi] Z^{-1} e^{-S[\phi]}$, where $S$ is the lattice-discretization of the continuous Euclidean action $\mathcal S$ of some quantum field theory. Since such densities arise as the approximation of the underlying functional density $[\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}$, we frame the task as an instance of operator learning. In particular, we propose to approximate a time-dependent operator $\mathcal V_t$ whose time integral provides a mapping between the functional distributions of the free theory $[\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]}$ and of the target theory $[\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}$. Whenever a particular lattice is chosen, the operator $\mathcal V_t$ can be discretized to a finite dimensional, time-dependent vector field $V_t$ 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 $[d\phi] Z_0^{-1} e^{-S_{0}[\phi]}$, $[d\phi] Z^{-1}e^{-S[\phi]}$. We run experiments on the $\phi^4$-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 speedup over training only a target lattice size.

We propose a novel machine learning method for sampling from the high-dimensional probability distributions of Lattice Field Theories, which is based on a single neural ODE layer and incorporates the full symmetries of the problem. We test our model on the $\phi^4$ theory, showing that it systematically outperforms previously proposed flow-based methods in sampling efficiency, and the improvement is especially pronounced for larger lattices. Furthermore, we demonstrate that our model can learn a continuous family of theories at once, and the results of learning can be transferred to larger lattices. Such generalizations further accentuate the advantages of machine learning methods.

Recently, machine learning algorithms have been used remarkably to study the equilibrium phase transitions, however there are only a few works have been done using this technique in the nonequilibrium phase transitions. In this work, we use the supervised learning with the convolutional neural network (CNN) algorithm and unsupervised learning with the density-based spatial clustering of applications with noise (DBSCAN) algorithm to study the nonequilibrium phase transition in two models. We use CNN and DBSCAN in order to determine the critical points for directed bond percolation (bond DP) model and Domany-Kinzel cellular automaton (DK) model. Both models have been proven to have a nonequilibrium phase transition belongs to the directed percolation (DP) universality class. In the case of supervised learning we train CNN using the images which are generated from Monte Carlo simulations of directed bond percolation. We use that trained CNN in studding the phase transition for the two models. In the case of unsupervised learning, we train DBSCAN using the raw data of Monte Carlo simulations. In this case, we retrain DBSCAN at each time we change the model or lattice size. Our results from both algorithms show that, even for a very small values of lattice size, machine can predict the critical points accurately for both models. Finally, we mention to that, the value of the critical point we find here for bond DP model using CNN or DBSCAN is exactly the same value that has been found using transfer learning with a domain adversarial neural network (DANN) algorithm.

In this paper with study phase transitions of the $q$-state Potts model, through a number of unsupervised machine learning techniques, namely Principal Component Analysis (PCA), $k$-means clustering, Uniform Manifold Approximation and Projection (UMAP), and Topological Data Analysis (TDA). Even though in all cases we are able to retrieve the correct critical temperatures $T_c(q)$, for $q = 3, 4$ and $5$, results show that non-linear methods as UMAP and TDA are less dependent on finite size effects, while still being able to distinguish between first and second order phase transitions. This study may be considered as a benchmark for the use of different unsupervised machine learning algorithms in the investigation of phase transitions.

The crucial role played by the underlying symmetries of high energy physics and lattice field theories calls for the implementation of such symmetries in the neural network architectures that are applied to the physical system under consideration. In these proceedings, we focus on the consequences of incorporating translational equivariance among the network properties, particularly in terms of performance and generalization. The benefits of equivariant networks are exemplified by studying a complex scalar field theory, on which various regression and classification tasks are examined. For a meaningful comparison, promising equivariant and non-equivariant architectures are identified by means of a systematic search. The results indicate that in most of the tasks our best equivariant architectures can perform and generalize significantly better than their non-equivariant counterparts, which applies not only to physical parameters beyond those represented in the training set, but also to different lattice sizes.

In recent years, the use of machine learning has become increasingly popular in the context of lattice field theories. An essential element of such theories is represented by symmetries, whose inclusion in the neural network properties can lead to high reward in terms of performance and generalizability. A fundamental symmetry that usually characterizes physical systems on a lattice with periodic boundary conditions is equivariance under spacetime translations. Here we investigate the advantages of adopting translationally equivariant neural networks in favor of non-equivariant ones. The system we consider is a complex scalar field with quartic interaction on a two-dimensional lattice in the flux representation, on which the networks carry out various regression and classification tasks. Promising equivariant and non-equivariant architectures are identified with a systematic search. We demonstrate that in most of these tasks our best equivariant architectures can perform and generalize significantly better than their non-equivariant counterparts, which applies not only to physical parameters beyond those represented in the training set, but also to different lattice sizes.