Alexander Hvatov

Mikhail Maslyaev, Alexander Hvatov

Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions than conventional approaches, such as sparse symbolic regression over the complete possible terms library. The equation discovery field contains two independent directions. The first one is purely mathematical and concerns differentiation, the object of optimization and its relation to the functional spaces and others. The second one is dedicated purely to the optimizational problem statement. Both topics are worth investigating to improve the algorithm's ability to handle experimental data a more artificial intelligence way, without significant pre-processing and a priori knowledge of their nature. In the paper, we consider the prevalence of either single-objective optimization, which considers only the discrepancy between selected terms in the equation, or multi-objective optimization, which additionally takes into account the complexity of the obtained equation. The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation.

Alexander Hvatov, Tatiana Tikhonova

The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change.

Alexander Hvatov, Roman Titov

Differential equation discovery, a machine learning subfield, is used to develop interpretable models, particularly in nature-related applications. By expertly incorporating the general parametric form of the equation of motion and appropriate differential terms, algorithms can autonomously uncover equations from data. This paper explores the prerequisites and tools for independent equation discovery without expert input, eliminating the need for equation form assumptions. We focus on addressing the challenge of assessing the adequacy of discovered equations when the correct equation is unknown, with the aim of providing insights for reliable equation discovery without prior knowledge of the equation form.

Yakov Golovanev, Alexander Hvatov

Most machine learning methods are used as a black box for modelling. We may try to extract some knowledge from physics-based training methods, such as neural ODE (ordinary differential equation). Neural ODE has advantages like a possibly higher class of represented functions, the extended interpretability compared to black-box machine learning models, ability to describe both trend and local behaviour. Such advantages are especially critical for time series with complicated trends. However, the known drawback is the high training time compared to the autoregressive models and long-short term memory (LSTM) networks widely used for data-driven time series modelling. Therefore, we should be able to balance interpretability and training time to apply neural ODE in practice. The paper shows that modern neural ODE cannot be reduced to simpler models for time-series modelling applications. The complexity of neural ODE is compared to or exceeds the conventional time-series modelling tools. The only interpretation that could be extracted is the eigenspace of the operator, which is an ill-posed problem for a large system. Spectra could be extracted using different classical analysis methods that do not have the drawback of extended time. Consequently, we reduce the neural ODE to a simpler linear form and propose a new view on time-series modelling using combined neural networks and an ODE system approach.

Mikhail Masliaev, Dmitry Gusarov, Ilya Markov et al.

Although neural operators are widely used in data-driven physical simulations, their training remains computationally expensive. Recent advances address this issue via downstream learning, where a model pretrained on simpler problems is fine-tuned on more complex ones. In this research, we investigate transformer-based neural operators, which have previously been applied only to specific problems, in a more general transfer learning setting. We evaluate their performance across diverse PDE problems, including extrapolation to unseen parameters, incorporation of new variables, and transfer from multi-equation datasets. Our results demonstrate that advanced neural operator architectures can effectively transfer knowledge across PDE problems.

Mikhail Masliaev, Ilya Markov, Alexander Hvatov

This paper explores the critical role of differentiation approaches for data-driven differential equation discovery. Accurate derivatives of the input data are essential for reliable algorithmic operation, particularly in real-world scenarios where measurement quality is inevitably compromised. We propose alternatives to the commonly used finite differences-based method, notorious for its instability in the presence of noise, which can exacerbate random errors in the data. Our analysis covers four distinct methods: Savitzky-Golay filtering, spectral differentiation, smoothing based on artificial neural networks, and the regularization of derivative variation. We evaluate these methods in terms of applicability to problems, similar to the real ones, and their ability to ensure the convergence of equation discovery algorithms, providing valuable insights for robust modeling of real-world processes.

Maria Khilchuk, Vladimir Latypov, Pavel Kleshchev et al.

Diffusion and Schrödinger Bridge models have established state-of-the-art performance in generative modeling but are often hampered by significant computational costs and complex training procedures. While continuous-time bridges promise faster sampling, overparameterized neural networks describe their optimal dynamics, and the underlying stochastic differential equations can be difficult to integrate efficiently. This work introduces a novel paradigm that uses surrogate models to create simpler, faster, and more flexible approximations of these dynamics. We propose two specific algorithms: SINDy Flow Matching (SINDy-FM), which leverages sparse regression to identify interpretable, symbolic differential equations from data, and a Neural-ODE reformulation of the Schrödinger Bridge (DSBM-NeuralODE) for flexible continuous-time parameterization. Our experiments on Gaussian transport tasks and MNIST latent translation demonstrate that these surrogates achieve competitive performance while offering dramatic improvements in efficiency and interpretability. The symbolic SINDy-FM models, in particular, reduce parameter counts by several orders of magnitude and enable near-instantaneous inference, paving the way for a new class of tractable and high-performing bridge models for practical deployment.

Elizaveta Ivanchik, Alexander Hvatov

In differential equation discovery algorithms, a priori expert knowledge is mainly used implicitly to constrain the form of the expected equation, making it impossible for the algorithm to truly discover equations. Instead, most differential equation discovery algorithms try to recover the coefficients for a known structure. In this paper, we describe an algorithm that allows the discovery of unknown equations using automatically or manually extracted background knowledge. Instead of imposing rigid constraints, we modify the structure space so that certain terms are likely to appear within the crossover and mutation operators. In this way, we mimic expertly chosen terms while preserving the possibility of obtaining any equation form. The paper shows that the extraction and use of knowledge allows it to outperform the SINDy algorithm in terms of search stability and robustness. Synthetic examples are given for Burgers, wave, and Korteweg--De Vries equations.

Mikhail Maslyaev, Alexander Hvatov

Equation discovery methods hold promise for extracting knowledge from physics-related data. However, existing approaches often require substantial prior information that significantly reduces the amount of knowledge extracted. In this paper, we enhance the EPDE algorithm -- an evolutionary optimization-based discovery framework. In contrast to methods like SINDy, which rely on pre-defined libraries of terms and linearities, our approach generates terms using fundamental building blocks such as elementary functions and individual differentials. Within evolutionary optimization, we may improve the computation of the fitness function as is done in gradient methods and enhance the optimization algorithm itself. By incorporating multi-objective optimization, we effectively explore the search space, yielding more robust equation extraction, even when dealing with complex experimental data. We validate our algorithm's noise resilience and overall performance by comparing its results with those from the state-of-the-art equation discovery framework SINDy.

Alexander Hvatov, Mikhail Maslyaev, Iana S. Polonskaya et al.

In modern data science, it is often not enough to obtain only a data-driven model with a good prediction quality. On the contrary, it is more interesting to understand the properties of the model, which parts could be replaced to obtain better results. Such questions are unified under machine learning interpretability questions, which could be considered one of the area's raising topics. In the paper, we use multi-objective evolutionary optimization for composite data-driven model learning to obtain the algorithm's desired properties. It means that whereas one of the apparent objectives is precision, the other could be chosen as the complexity of the model, robustness, and many others. The method application is shown on examples of multi-objective learning of composite models, differential equations, and closed-form algebraic expressions are unified and form approach for model-agnostic learning of the interpretable models.

Mikhail Maslyaev, Alexander Hvatov

Usually, the systems of partial differential equations (PDEs) are discovered from observational data in the single vector equation form. However, this approach restricts the application to the real cases, where, for example, the form of the external forcing is of interest. In the paper, a multi-objective co-evolution algorithm is described. The single equations within the system and the system itself are evolved simultaneously to obtain the system. This approach allows discovering the systems with the form-independent equations. In contrast to the single vector equation, a component-wise system is more suitable for expert interpretation and, therefore, for applications. The example of the two-dimensional Navier-Stokes equation is considered.

Nikolay O. Nikitin, Ilia Revin, Alexander Hvatov et al.

The paper describes the usage of intelligent approaches for field development tasks that may assist a decision-making process. We focused on the problem of wells location optimization and two tasks within it: improving the quality of oil production estimation and estimation of reservoir characteristics for appropriate wells allocation and parametrization, using machine learning methods. For oil production estimation, we implemented and investigated the quality of forecasting models: physics-based, pure data-driven, and hybrid one. The CRMIP model was chosen as a physics-based approach. We compare it with the machine learning and hybrid methods in a frame of oil production forecasting task. In the investigation of reservoir characteristics for wells location choice, we automated the seismic analysis using evolutionary identification of convolutional neural network for the reservoir detection. The Volve oil field dataset was used as a case study to conduct the experiments. The implemented approaches can be used to analyze different oil fields or adapted to similar physics-related problems.

Alexander Hvatov, Mikhail Maslyaev

The modern machine learning methods allow one to obtain the data-driven models in various ways. However, the more complex the model is, the harder it is to interpret. In the paper, we describe the algorithm for the mathematical equations discovery from the given observations data. The algorithm combines genetic programming with the sparse regression. This algorithm allows obtaining different forms of the resulting models. As an example, it could be used for governing analytical equation discovery as well as for partial differential equations (PDE) discovery. The main idea is to collect a bag of the building blocks (it may be simple functions or their derivatives of arbitrary order) and consequently take them from the bag to create combinations, which will represent terms of the final equation. The selected terms pass to the evolutionary algorithm, which is used to evolve the selection. The evolutionary steps are combined with the sparse regression to pick only the significant terms. As a result, we obtain a short and interpretable expression that describes the physical process that lies beyond the data. In the paper, two examples of the algorithm application are described: the PDE discovery for the metocean processes and the function discovery for the acoustics.

Michail Maslyaev, Alexander Hvatov, Anna Kalyuzhnaya

The data-driven models allow one to define the model structure in cases when a priori information is not sufficient to build other types of models. The possible way to obtain physical interpretation is the data-driven differential equation discovery techniques. The existing methods of PDE (partial derivative equations) discovery are bound with the sparse regression. However, sparse regression is restricting the resulting model form, since the terms for PDE are defined before regression. The evolutionary approach described in the article has a symbolic regression as the background instead and thus has fewer restrictions on the PDE form. The evolutionary method of PDE discovery (EPDE) is described and tested on several canonical PDEs. The question of robustness is examined on a noised data example.

Alexander Hvatov, Nikolay O. Nikitin, Anna V. Kalyuzhnaya et al.

High-resolution regional hindcasting of ocean and sea ice plays an important role in the assessment of shipping and operational risks in the Arctic Ocean. The ice-ocean model NEMO-LIM3 was modified to improve its simulation quality for appropriate spatio-temporal resolutions. A multigrid model setup with connected coarse- (14 km) and fine-resolution (5 km) model configurations was devised. These two configurations were implemented and run separately. The resulting computational cost was lower when compared to that of the built-in AGRIF nesting system. Ice and tracer boundary-condition schemes were modified to achieve the correct interaction between coarse- and fine grids through a long ice-covered open boundary. An ice-restoring scheme was implemented to reduce spin-up time. The NEMO-LIM3 configuration described in this article provides more flexible and customisable tools for high-resolution regional Arctic simulations.