The paper considers the question of the possibility of recovering symmetric stable states of a bistable medium in the inverse problem for a nonlinear singularly perturbed autowave equation by data given on the position of an autowave front propagating through it. It is shown that under certain conditions, this statement of the problem is ill-posed in the sense of the non-uniqueness of the solution. A regularizing approach to its solution was proposed.

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

The paper proposes a new two-level neural network approach to joint inversion of gravitational and magnetic fields in the problem of mineral exploration. The lower level uses two neural networks with equivalent architecture. The first network calculates the distribution of gravitational field sources in a given domain based on measurements of this field on a certain remote surface. The second one finds the distribution of magnetic field sources in the same area based on magnetic field measurements. At the upper level, a structural residual between the two obtained source distributions is calculated, which determines the “degree of difference” of their geometries. The resulting neural network is trained to minimize this structural discrepancy. As a result, this two-level system implements a computational algorithm that finds geometrically close sources of fields of different nature. Test simulations demonstrate the high quality of joint inversion of model gravitational and magnetic fields within the proposed aproach. В работе предлагается новый двухуровневый нейросетевой подход к совместной инверсии гравитационных и магнитных полей в адаче поиска полезных ископаемых. Нижний уровень использует две эквивалентные по архитектуре нейронные сети. Одна вычисляет распределение источников гравитационного поля в заданной области по измерениям этого поля на некоторой удаленной поверхности. Вторая находит в той же области распределение источников магнитного поля по измерениям магнитного поля. На верхнем уровне вычисляется структурная невязка двух полученных распределений источников, которая определяет “степень различия” их геометрий. Результирующая нейронная сеть обучается таким образом, чтобы структурная невязка была минимальной. В итоге эта двухуровневая система реализует вычислительный алгоритм, который находит геометрически близкие источники полей различной природы. Тестовые расчеты демонстрируют высокое качество совместной инверсии модельных гравитационных и магнитных полей в рамках предложенного подхода.

We consider the optimization of a neural network previously developed by the authors for the joint inversion of 3D gravitational and magnetic fields in the context of mineral exploration. The distinctive feature of this neural network is that it solves ill-posed (ill-conditioned) inverse problems. The neural network implements a special two-level algorithm. The lower level of the algorithm uses two neural networks with equivalent architectures. The first of them computes the gravitational field sources in a given domain from measurements of this field on a remote surface. The second neural network processes magnetic field measured on the same surface to find magnetic sources in the same domain. The found source distributions are used at the upper level of the algorithm to calculate their structural residual, which determines the degree of difference (closeness) of their geometries. As a result, minimizing this residual, when training a neural network at the upper level, implements a computational algorithm that yields geometrically close source distributions of different fields. The article examines in detail the possibilities of optimizing some elements of the neural networks and the algorithms used (datasets, training process, specific form of loss functions, etc.) Test calculations for model problem demonstrate high quality of joint inversion by our optimized neural networks approach. Calculations were also carried out for the joint processing of real-feald data from gravity and magnetic exploration in Jussara region, Goias State, Brazil. The article also considers the issue of determining in joint field inversion not only the geometric distribution of sources, but also their physical intensities.