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Source analysis of magnetoencephalography (MEG) data requires the computation of the magnetic fields induced by current sources in the brain. This so‐called MEG forward problem includes an accurate estimation of the volume conduction effects in the human head. Here, we introduce the Cut finite element method (CutFEM) for the MEG forward problem. CutFEM's meshing process imposes fewer restrictions on tissue anatomy than tetrahedral meshes while being able to mesh curved geometries contrary to hexahedral meshing. To evaluate the new approach, we compare CutFEM with a boundary element method (BEM) that distinguishes three tissue compartments and a 6‐compartment hexahedral FEM in an n = 19 group study of somatosensory evoked fields (SEF). The neural generators of the 20 ms post‐stimulus SEF components (M20) are reconstructed using both an unregularized and a regularized inversion approach. Changing the forward model resulted in reconstruction differences of about 1 centimeter in location and considerable differences in orientation. The tested 6‐compartment FEM approaches significantly increase the goodness of fit to the measured data compared with the 3‐compartment BEM. They also demonstrate higher quasi‐radial contributions for sources below the gyral crowns. Furthermore, CutFEM improves source separability compared with both other approaches. We conclude that head models with 6 compartments rather than 3 and the new CutFEM approach are valuable additions to MEG source reconstruction, in particular for sources that are predominantly radial. CutFEM is a new magnetoencephalography (MEG) forward modeling technique, facilitating the mesh generation process. CutFEM enhances sensitivity to quasi‐radial source contributions and improves the ability to distinguish proximate neural sources compared to two competing boundary and finite element approaches.

We consider an inverse problem of recovering the mortality rate in the honey bee difference equation model, that tracks a forage honeybee leaving and entering the hive each day. We concentrate our analysis to the model without pesticide contamination in the symmetric spatial environment. Thus, the mathematical problem is formulated as a symmetric inverse problem for reaction coefficient at final time constraint. We use the overspecified information to transform the inverse coefficient problem to the forward problem with non-local terms in the differential operator and the initial condition. First, we apply semidiscretization in space to the new nonsymmetric differential operator. Then, the resulting non-local nonsymmetric system of ordinary differential equations (ODEs) is discretized by three iterative numerical schemes using different time stepping. Results of numerical experiments which compare the efficiency of the numerical schemes are discussed. Results from numerical tests with synthetic and real data are presented and discussed, as well.

This article describes a trilinear immersed finite element method for solving the electroencephalography forward problem, which is a three-dimensional elliptic interface problem in the head geometry. The method uses hexahedral Cartesian meshes independent of the interfaces between head tissues, thus avoiding the sometimes difficult task of generating geometry fitting meshes. The interfaces are represented with level sets, and the finite element space is locally modified to better approximate the discontinuities of the solution. Numerical results show that this method achieves the same accuracy as the standard linear finite element method with geometry fitting mesh. [PUBLICATION ABSTRACT]

This paper mainly introduces the parallel physics-informed neural networks (PPINNs) method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries (VC-MKdV) equation. For the forward problem of the VC-MKdV equation, the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed. Furthermore, the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation. As for the data-driven inverse problem of the VC-MKdV equation, the author introduces a parallel neural networks to separately train the solution function and coefficient function, successfully addressing the function discovery problem of the VC-MKdV equation. To further enhance the network’s generalization ability and noise robustness, the author incorporates two regularization strategies into the PPINNs. An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation, and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.

To study the feasibility of using machine learning technology to solve the forward problem (prediction of aerodynamic parameters) and the inverse problem (prediction of geometric parameters) of turbine blades, this paper built a forward problem model based on backpropagation artificial neural networks (BP-ANNs) and an inverse problem model based on radial basis function artificial neural networks (RBF-ANNs). The S2 (a stream surface obtained by extending a radial curve in turbo blades) calculation program was used to generate the dataset for single-stage turbo blades, and the back propagation algorithm was used to train the model. The parameters of five blade sections in a single-stage turbine were selected as inputs of the forward problem model, including stagger angle, inlet geometric angle, outlet geometric angle, wedge angle of leading edge pressure side, wedge angle of leading edge suction side, wedge angle of trailing edge, rear bending angle, and leading edge diameter. The outputs are efficiency, power, mass flow, relative exit Mach number, absolute exit Mach number, relative exit flow angle, absolute exit flow angle and reaction degree, which are eight aerodynamic parameters. The inputs and outputs of the inverse problem model are the opposite of that of the forward problem model. The models can accurately predict the aerodynamic parameters and geometric parameters, and the mean square errors (MSEs) of the forward problem test set and the inverse problem test set are 0.001 and 0.000 35, respectively. This study shows that machine learning technology based on neural networks can be flexibly applied to the design of forward and inverse problems of turbine blades, and the models built by this method have practical application value in regression prediction problems.

Understanding the complex patterns in space–time exhibited by active systems has been the subject of much interest in recent times. Complementing this forward problem is the inverse problem of controlling active matter. Here, we use optimal control theory to pose the problem of transporting a slender drop of an active fluid and determine the dynamical profile of the active stresses to move it with minimal viscous dissipation. By parametrizing the position and size of the drop using a low-order description based on lubrication theory, we uncover a natural “gather–move–spread” strategy that leads to an optimal bound on the maximum achievable displacement of the drop relative to its size. In the continuum setting, the competition between passive surface tension and active controls generates richer behavior with futile oscillations and complex drop morphologies that trade internal dissipation against the transport cost to select optimal strategies. Our work combines active hydrodynamics and optimal control in a tractable and interpretable framework and begins to pave the way for the spatiotemporal manipulation of active matter.

Computer-aided engineering (CAE) is now an essential instrument that aids in engineering decision-making. Statistical model calibration and validation has recently drawn great attention in the engineering community for its applications in practical CAE models. The objective of this paper is to review the state-of-the-art and trends in statistical model calibration and validation, based on the available extensive literature, from the perspective of uncertainty structures. After a brief discussion about uncertainties, this paper examines three problem categories—the forward problem, the inverse problem, and the validation problem—in the context of techniques and applications for statistical model calibration and validation.

To enhance the precision and efficiency of result prediction, we proposed a parallel hard-constraint physics-informed neural networks (phPINN) by combining the parallel fully-connected neural network structure and the residual-based adaptive refinement method. We discussed the forward and inverse problems of the nonlinear Schrödinger–Maxwell–Bloch equation via the phPINN. In the forward problem, we predict five forms of soliton solutions and rogue wave dynamics under corresponding initial and boundary conditions; In the inverse problem, we predict the equation parameter using the training data with different noise intensities, initial values, and solution forms. The predicted parameters achieve a relative error of less than 1%. These results validate the effectiveness of the phPINN algorithm in solving forward and inverse problems of three-component coupled equations.

Modeling water flow in unsaturated soils is vital for describing various hydrological and ecological phenomena. Soil water dynamics is described by well-established physical laws (Richardson–Richards equation – RRE). Solving the RRE is difficult due to the inherent nonlinearity of the processes, and various numerical methods have been proposed to solve the issue. However, applying the methods to practical situations is very challenging because they require well-defined initial and boundary conditions. Recent advances in machine learning and the growing availability of soil moisture data provide new opportunities for addressing the lingering challenges. Specifically, physics-informed machine learning allows both the known physics and data-driven modeling to be taken advantage of. Here, we present a physics-informed neural network (PINN) method that approximates the solution to the RRE using neural networks while concurrently matching available soil moisture data. Although the ability of PINNs to solve partial differential equations, including the RRE, has been demonstrated previously, its potential applications and limitations are not fully known. This study conducted a comprehensive analysis of PINNs and carefully tested the accuracy of the solutions by comparing them with analytical solutions and accepted traditional numerical solutions. We demonstrated that the solutions by PINNs with adaptive activation functions are comparable with those by traditional methods. Furthermore, while a single neural network (NN) is adequate to represent a homogeneous soil, we showed that soil moisture dynamics in layered soils with discontinuous hydraulic conductivities are correctly simulated by PINNs with domain decomposition (using separate NNs for each unique layer). A key advantage of PINNs is the absence of the strict requirement for precisely prescribed initial and boundary conditions. In addition, unlike traditional numerical methods, PINNs provide an inverse solution without repeatedly solving the forward problem. We demonstrated the application of these advantages by successfully simulating infiltration and redistribution constrained by sparse soil moisture measurements. As a free by-product, we gain knowledge of the water flux over the entire flow domain, including the unspecified upper and bottom boundary conditions. Nevertheless, there remain challenges that require further development. Chiefly, PINNs are sensitive to the initialization of NNs and are significantly slower than traditional numerical methods.

Physics-informed neural networks (PINNs) have shown remarkable prospects in solving the forward and inverse problems involving partial differential equations (PDEs). The method embeds PDEs into the neural network by calculating the PDE loss at a set of collocation points, providing advantages such as meshfree and more convenient adaptive sampling. However, when solving PDEs using nonuniform collocation points, PINNs still face challenge regarding inefficient convergence of PDE residuals or even failure. In this work, we first analyze the ill-conditioning of the PDE loss in PINNs under nonuniform collocation points. To address the issue, we define volume weighting residual and propose volume weighting physics-informed neural networks (VW-PINNs). Through weighting the PDE residuals by the volume that the collocation points occupy within the computational domain, we embed explicitly the distribution characteristics of collocation points in the loss evaluation. The fast and sufficient convergence of the PDE residuals for the problems involving nonuniform collocation points is guaranteed. Considering the meshfree characteristics of VW-PINNs, we also develop a volume approximation algorithm based on kernel density estimation to calculate the volume of the collocation points. We validate the universality of VW-PINNs by solving the forward problems involving flow over a circular cylinder and flow over the NACA0012 airfoil under different inflow conditions, where conventional PINNs fail. By solving the Burgers’ equation, we verify that VW-PINNs can enhance the efficiency of existing the adaptive sampling method in solving the forward problem by three times, and can reduce the relative L 2 error of conventional PINNs in solving the inverse problem by more than one order of magnitude.