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Physics-informed neural networks (PINNs) have shown remarkable prospects in the solving the forward and inverse problems involving partial differential equations (PDEs). The method embeds PDEs into the neural network by calculating PDE loss at a series 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-weighted residual and propose volume-weighted 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 spatial distribution characteristics of collocation points in the residual 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 verify 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 3 times, and can reduce the relative error of conventional PINNs in solving the inverse problem by more than one order of magnitude.

Automated medical image segmentation plays an important role in many clinical applications, which however is a very challenging task, due to complex background texture, lack of clear boundary and significant shape and texture variation between images. Many researchers proposed an encoder-decoder architecture with skip connections to combine low-level feature maps from the encoder path with high-level feature maps from the decoder path for automatically segmenting medical images. The skip connections have been shown to be effective in recovering fine-grained details of the target objects and may facilitate the gradient back-propagation. However, not all the feature maps transmitted by those connections contribute positively to the network performance. In this paper, to adaptively select useful information to pass through those skip connections, we propose a novel 3D network with self-supervised function, named selective information passing network (SIP-Net). We evaluate our proposed model on the MICCAI Prostate MR Image Segmentation 2012 Grant Challenge dataset, TCIA Pancreas CT-82 and MICCAI 2017 Liver Tumor Segmentation (LiTS) Challenge dataset. The experimental results across these data sets show that our model achieved improved segmentation results and outperformed other state-of-the-art methods. The source code of this work is available at https://github.com/ahukui/SIPNet.

A dynamic method to solve the Non-linear Programming (NLP) problem with Equality Constraints (ECs) and Inequality Constraints (IECs) is proposed. Inspired by the Lyapunov continuous-time dynamics stability theory in the control field, the optimal solution is analogized to the stable equilibrium point of a finite-dimensional dynamic system and it is solved in an asymptotic manner. Under the premise that the Karush-Kuhn-Tucker (KKT) optimality condition exists, the Dynamic Optimization Equation (DOE), which has the same dimension to that of the optimization parameter vector, is established and its solution will converge to the optimal solution of the NLP globally with a theoretical guarantee. Using the matrix pseudo-inverse, the DOE is valid even without the linearly independent regularity requirement on the nonlinear constraints. In addition, the analytic expressions of the Lagrange multipliers and KKT multipliers, which adjoin the ECs and the IECs respectively during the entire optimization process, are also derived. Via the proposed method, the NLP may be transformed to the Initial-value Problem (IVP) to be solved, with mature Ordinary Differential Equation (ODE) integration methods. Illustrative examples are solved and it is shown that the dynamic method developed may produce the right numerical solutions with high efficiency.

The first evolution equation is derived under the Variation Evolving Method (VEM) that seeks optimal solutions with the variation evolution principle. To improve the performance, its compact form is developed. By replacing the states and costates variation evolution with that of the controls, the dimension-reduced Evolution Partial Differential Equation (EPDE) only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary. With this equation, the scale of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is significantly reduced. Illustrative examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision, and the efficiency may be higher for the dense discretization. Moreover, in discussing the connections to the classic iteration methods, it is uncovered that the computation scheme of the gradient method is the discrete implementation of the third evolution equation, and the compact form of the first evolution equation is a continuous realization of the Newton type iteration mechanism., Comment: arXiv admin note: substantial text overlap with arXiv:1802.04663, arXiv:1801.10486, arXiv:1801.01383, arXiv:1802.02140, arXiv:1801.07395

The Variation Evolving Method (VEM) that originates from the continuous-time dynamics stability theory seeks the optimal solutions with variation evolution principle. After establishing the first and the second evolution equations within its frame, the third evolution equation is developed. This equation only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary since the equation can eliminate possible infeasibilities. With this equation, the dimension of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is greatly reduced. Therefore it might relieve the computation burden in seeking solutions. Illustrative examples are solved and it is shown that the proposed equation may produce more precise numerical solutions than the second evolution equation, and its computation time may be shorter for the dense discretization., Comment: Key words: Optimal control, dynamics stability, variation evolution, evolution partial differential equation, the third evolution equation, initial-value problem. arXiv admin note: substantial text overlap with arXiv:1801.10486, arXiv:1801.01383, arXiv:1801.07395, arXiv:1712.09702 and text overlap with arXiv:1802.02140, arXiv:1709.02242

The Variation Evolving Method (VEM), which seeks the optimal solutions with the variation evolution principle, is further developed to be more flexible in solving the Optimal Control Problems (OCPs) with terminal constraint. With the first-order stable dynamics to eliminate the infeasibilities, the Modified Evolution Partial Differential Equation (MEPDE) that is valid in the infeasible solution domain is proposed, and a Lyapunov functional is constructed to theoretically ensure its validity. In particular, it is proved that even with the infinite-time convergence dynamics, the violated terminal inequality constraints, which are inactive for the optimal solution, will enter the feasible domain in finite time. Through transforming the MEPDE to the finite-dimensional Initial-value Problem (IVP) with the semi-discrete method, the OCPs may be solved with common Ordinary Differential Equation (ODE) numerical integration methods. Illustrative examples are presented to show the effectiveness of the proposed method., Comment: arXiv admin note: substantial text overlap with arXiv:1801.07395, arXiv:1801.01383; text overlap with arXiv:1712.09702, arXiv:1709.02242

Enlightened from the inverse consideration of the stable continuous-time dynamics evolution, the Variation Evolving Method (VEM) analogizes the optimal solution to the equilibrium point of an infinite-dimensional dynamic system and solves it in an asymptotically evolving way. In this paper, the compact version of the VEM is further developed for the computation of Optimal Control Problems (OCPs) with terminal constraint. The corresponding Evolution Partial Differential Equation (EPDE), which describes the variation motion towards the optimal solution, is derived, and the costate-free optimality conditions are established. The explicit analytic expressions of the costates and the Lagrange multipliers adjoining the terminal constraint, related to the states and the control variables, are presented. With the semi-discrete method in the field of PDE numerical calculation, the EPDE is discretized as finite-dimensional Initial-value Problems (IVPs) to be solved, with common Ordinary Differential Equation (ODE) numerical integration methods., Comment: arXiv admin note: substantial text overlap with arXiv:1709.02242, arXiv:1712.09702, arXiv:1711.02998, arXiv:1703.10263

The feedback linearization method is further developed for the controller design on general nonlinear systems. Through the Lyapunov stability theory, the intractable nonlinear implicit algebraic control equations are effectively solved, and the asymptotically tracking performance is guaranteed. Moreover, it is proved that the controller may be used in an inverse-free version to the set-point control. With this method, a nonlinear aircraft outer-loop trajectory controller is developed. For the concern regarding the controller's robustness, the integral control technique is combined to counteract the adverse effect from modeling errors. Simulation results verify the well performance of the proposed controller.

A compact version of the variation evolving method (VEM) is developed in the primal variable space for optimal control computation. Following the idea that originates from the Lyapunov continuous-time dynamics stability theory in the control field, the optimal solution is analogized to the stable equilibrium point of a dynamic system and obtained asymptotically through the variation motion. With the introduction of a virtual dimension, namely the variation time, the evolution partial differential equation (EPDE), which seeks the optimal solution with a theoretical guarantee, is developed for the optimal control problem (OCP) with free terminal states, and the equivalent optimality conditions with no employment of costates are established in the primal space. These conditions show that the optimal feedback control law is generally not analytically available because the optimal control is related to the future states. Since the derived EPDE is suitable to be computed with the semi-discrete method in the field of PDE numerical calculation, the optimal solution may be obtained by solving the resulting finite-dimensional initial-value problem (IVP)., Comment: 22 pages, 8 figures. arXiv admin note: text overlap with arXiv:1703.10263