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We introduce DynAMO, a reinforcement learning paradigm for Dynamic Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool for optimizing computational cost and solution accuracy in numerical methods for partial differential equations. However, traditional adaptive mesh refinement approaches for time-dependent problems typically rely only on instantaneous error indicators to guide adaptivity. As a result, standard strategies often require frequent remeshing to maintain accuracy. In the DynAMO approach, multi-agent reinforcement learning is used to discover new local refinement policies that can anticipate and respond to future solution states by producing meshes that deliver more accurate solutions for longer time intervals. By applying DynAMO to discontinuous Galerkin methods for the linear advection and compressible Euler equations in two dimensions, we demonstrate that this new mesh refinement paradigm can outperform conventional threshold-based strategies while also generalizing to different mesh sizes, remeshing and simulation times, and initial conditions.

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such `gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by non-causal dual problems.

What can be learned about causality and experimentation from passive data? This question is salient given recent successes of passively-trained language models in interactive domains such as tool use. Passive learning is inherently limited. However, we show that purely passive learning can in fact allow an agent to learn generalizable strategies for determining and using causal structures, as long as the agent can intervene at test time. We formally illustrate that learning a strategy of first experimenting, then seeking goals, can allow generalization from passive learning in principle. We then show empirically that agents trained via imitation on expert data can indeed generalize at test time to infer and use causal links which are never present in the training data; these agents can also generalize experimentation strategies to novel variable sets never observed in training. We then show that strategies for causal intervention and exploitation can be generalized from passive data even in a more complex environment with high-dimensional observations, with the support of natural language explanations. Explanations can even allow passive learners to generalize out-of-distribution from perfectly-confounded training data. Finally, we show that language models, trained only on passive next-word prediction, can generalize causal intervention strategies from a few-shot prompt containing examples of experimentation, together with explanations and reasoning. These results highlight the surprising power of passive learning of active causal strategies, and may help to understand the behaviors and capabilities of language models.

The maximum likelihood method is the best-known method for estimating the probabilities behind the data. However, the conventional method obtains the probability model closest to the empirical distribution, resulting in overfitting. Then regularization methods prevent the model from being excessively close to the wrong probability, but little is known systematically about their performance. The idea of regularization is similar to error-correcting codes, which obtain optimal decoding by mixing suboptimal solutions with an incorrectly received code. The optimal decoding in error-correcting codes is achieved based on gauge symmetry. We propose a theoretically guaranteed regularization in the maximum likelihood method by focusing on a gauge symmetry in Kullback -- Leibler divergence. In our approach, we obtain the optimal model without the need to search for hyperparameters frequently appearing in regularization.

Continual Learning (CL) is a process in which there is still huge gap between human and deep learning model efficiency. Recently, many CL algorithms were designed. Most of them have many problems with learning in dynamic and complex environments. In this work new architecture based approach Ada-QPacknet is described. It incorporates the pruning for extracting the sub-network for each task. The crucial aspect in architecture based CL methods is theirs capacity. In presented method the size of the model is reduced by efficient linear and nonlinear quantisation approach. The method reduces the bit-width of the weights format. The presented results shows that low bit quantisation achieves similar accuracy as floating-point sub-network on a well-know CL scenarios. To our knowledge it is the first CL strategy which incorporates both compression techniques pruning and quantisation for generating task sub-networks. The presented algorithm was tested on well-known episode combinations and compared with most popular algorithms. Results show that proposed approach outperforms most of the CL strategies in task and class incremental scenarios.

This work tackles the problem of finding a good ansatz initialization for Variational Quantum Algorithms (VQAs), by proposing CAFQA, a Clifford Ansatz For Quantum Accuracy. The CAFQA ansatz is a hardware-efficient circuit built with only Clifford gates. In this ansatz, the parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation. The resulting initial states always equal or outperform traditional classical initialization (e.g., Hartree-Fock), and enable high-accuracy VQA estimations. CAFQA is well-suited to classical computation because: a) Clifford-only quantum circuits can be exactly simulated classically in polynomial time, and b) the discrete Clifford space is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation (up to 18 qubits), CAFQA's Clifford Ansatz achieves a mean accuracy of nearly 99% and recovers as much as 99.99% of the molecular correlation energy that is lost in Hartree-Fock initialization. CAFQA achieves mean accuracy improvements of 6.4x and 56.8x, over the state-of-the-art, on different metrics. The scalability of the approach allows for preliminary ground state energy estimation of the challenging chromium dimer (Cr$_2$) molecule. With CAFQA's high-accuracy initialization, the convergence of VQAs is shown to accelerate by 2.5x, even for small molecules. Furthermore, preliminary exploration of allowing a limited number of non-Clifford (T) gates in the CAFQA framework, shows that as much as 99.9% of the correlation energy can be recovered at bond lengths for which Clifford-only CAFQA accuracy is relatively limited, while remaining classically simulable.

Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen's principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for overparameterized models. MDL-COMP is defined via an optimality criterion over the encodings induced by a good Ridge estimator class. We provide an extensive theoretical characterization of MDL-COMP for linear models and kernel methods and show that it is not just a function of parameter count, but rather a function of the singular values of the design or the kernel matrix and the signal-to-noise ratio. For a linear model with $n$ observations, $d$ parameters, and i.i.d. Gaussian predictors, MDL-COMP scales linearly with $d$ when $d<n$, but the scaling is exponentially smaller -- $\log d$ for $d>n$. For kernel methods, we show that MDL-COMP informs minimax in-sample error, and can decrease as the dimensionality of the input increases. We also prove that MDL-COMP upper bounds the in-sample mean squared error (MSE). Via an array of simulations and real-data experiments, we show that a data-driven Prac-MDL-COMP informs hyper-parameter tuning for optimizing test MSE with ridge regression in limited data settings, sometimes improving upon cross-validation and (always) saving computational costs. Finally, our findings also suggest that the recently observed double decent phenomenons in overparameterized models might be a consequence of the choice of non-ideal estimators.

Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD suit better for that of interior GSVD components.

In this paper, we propose to regularize ill-posed inverse problems using a deep hierarchical variational autoencoder (HVAE) as an image prior. The proposed method synthesizes the advantages of i) denoiser-based Plug \& Play approaches and ii) generative model based approaches to inverse problems. First, we exploit VAE properties to design an efficient algorithm that benefits from convergence guarantees of Plug-and-Play (PnP) methods. Second, our approach is not restricted to specialized datasets and the proposed PnP-HVAE model is able to solve image restoration problems on natural images of any size. Our experiments show that the proposed PnP-HVAE method is competitive with both SOTA denoiser-based PnP approaches, and other SOTA restoration methods based on generative models.

Lecture notes from the course given by Professor Sara A. Solla at the Les Houches summer school on "Statistical physics of Machine Learning". The notes discuss neural information processing through the lens of Statistical Physics. Contents include Bayesian inference and its connection to a Gibbs description of learning and generalization, Generalized Linear Models as a controlled alternative to backpropagation through time, and linear and non-linear techniques for dimensionality reduction.