Vladimir Fanaskov

Vladimir Fanaskov

We propose a convenient matrix-free neural architecture for the multigrid method. The architecture is simple enough to be implemented in less than fifty lines of code, yet it encompasses a large number of distinct multigrid solvers. We argue that a fixed neural network without dense layers can not realize an efficient iterative method. Because of that, standard training protocols do not lead to competitive solvers. To overcome this difficulty, we use parameter sharing and serialization of layers. The resulting network can be trained on linear problems with thousands of unknowns and retains its efficiency on problems with millions of unknowns. From the point of view of numerical linear algebra network's training corresponds to finding optimal smoothers for the geometric multigrid method. We demonstrate our approach on a few second-order elliptic equations. For tested linear systems, we obtain from two to five times smaller spectral radius of the error propagation matrix compare to a basic linear multigrid with Jacobi smoother.

Vladimir Fanaskov, Vladislav Trifonov, Alexander Rudikov et al.

It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of the problem. A practical way to apply such a scheme is to compute subspaces for a selected set of parameters in the computationally demanding offline stage and in the online stage approximate subspace for unknown parameters by interpolation. For realistic problems the space of parameters is high dimensional, which renders classical interpolation strategies infeasible or unreliable. We propose to relax the interpolation problem to regression, introduce several loss functions suitable for subspace data, and use a neural network as an approximation to high-dimensional target function. To further simplify a learning problem we introduce redundancy: in place of predicting subspace of a given dimension we predict larger subspace. We show theoretically that this strategy decreases the complexity of the mapping for elliptic eigenproblems with constant coefficients and makes the mapping smoother for general smooth function on the Grassmann manifold. Empirical results also show that accuracy significantly improves when larger-than-needed subspaces are predicted. With the set of numerical illustrations we demonstrate that subspace regression can be useful for a range of tasks including parametric eigenproblems, deflation techniques, relaxation methods, optimal control and solution of parametric partial differential equations.

Vladimir Fanaskov, Ivan Oseledets

Artificial Kuramoto oscillatory neurons were recently introduced as an alternative to threshold units. Empirical evidence suggests that oscillatory units outperform threshold units in several tasks including unsupervised object discovery and certain reasoning problems. The proposed coupling mechanism for these oscillatory neurons is heterogeneous, combining a generalized Kuramoto equation with standard coupling methods used for threshold units. In this research note, we present a theoretical framework that clearly distinguishes oscillatory neurons from threshold units and establishes a coupling mechanism between them. We argue that, from a biological standpoint, oscillatory and threshold units realise distinct aspects of neural coding: roughly, threshold units model intensity of neuron firing, while oscillatory units facilitate information exchange by frequency modulation. To derive interaction between these two types of units, we constrain their dynamics by focusing on dynamical systems that admit Lyapunov functions. For threshold units, this leads to Hopfield associative memory model, and for oscillatory units it yields a specific form of generalized Kuramoto model. The resulting dynamical systems can be naturally coupled to form a Hopfield-Kuramoto associative memory model, which also admits a Lyapunov function. Various forms of coupling are possible. Notably, oscillatory neurons can be employed to implement a low-rank correction to the weight matrix of a Hopfield network. This correction can be viewed either as a form of Hebbian learning or as a popular LoRA method used for fine-tuning of large language models. We demonstrate the practical realization of this particular coupling through illustrative toy experiments.

Vladimir Fanaskov, Tianchi Yu, Alexander Rudikov et al.

The primal approach to physics-informed learning is a residual minimization. We argue that residual is, at best, an indirect measure of the error of approximate solution and propose to train with error majorant instead. Since error majorant provides a direct upper bound on error, one can reliably estimate how close PiNN is to the exact solution and stop the optimization process when the desired accuracy is reached. We call loss function associated with error majorant \textbf{Astral}: neur\textbf{A}l a po\textbf{ST}erio\textbf{R}i function\textbf{A}l \textbf{L}oss. To compare Astral and residual loss functions, we illustrate how error majorants can be derived for various PDEs and conduct experiments with diffusion equations (including anisotropic and in the L-shaped domain), convection-diffusion equation, temporal discretization of Maxwell's equation, magnetostatics and nonlinear elastoplasticity problems. The results indicate that Astral loss is competitive to the residual loss, typically leading to faster convergence and lower error. The main benefit of using Astral loss comes from its ability to estimate error, which is impossible with other loss functions. Our experiments indicate that the error estimate obtained with Astral loss is usually tight enough, e.g., for a highly anisotropic equation, on average, Astral overestimates error by a factor of $1.5$, and for convection-diffusion by a factor of $1.7$. We further demonstrate that Astral loss is better correlated with error than residual and is a more reliable predictor of the error value. Moreover, unlike residual, the error indicator obtained from Astral loss has a superb spatial correlation with error. Backed with the empirical and theoretical results, we argue that one can productively use Astral loss to perform reliable error analysis and approximate PDE solutions with accuracy similar to standard residual-based techniques.