A discontinuity and cusp capturing PINN for Stokes interface problems with discontinuous viscosity and singular forces

In this paper, we present a discontinuity and cusp capturing physics-informed neural network (PINN) to solve Stokes equations with a piecewise-constant viscosity and singular force along an interface. We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface. Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface, we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity, respectively. The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs. These two augmented inputs provide the interface information, so we assume that a level set function is given and its zero level set indicates the position of the interface. The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity, while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation. We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature. Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods

A New Combined Boost Converter with Improved Voltage Gain as a Battery-Powered Front-End Interface for Automotive Audio Amplifiers

High boost DC/DC voltage conversion is always indispensable in a power electronic interface of certain battery-powered electrical equipment. However, a conventional boost converter works for a wide duty cycle for such high voltage gain, which increases power consumption and has low reliability problems. In order to solve this issue, a new battery-powered combined boost converter with an interleaved structure consisting of two phases used in automotive audio amplifier is presented. The first phase uses a conventional boost converter; the second phase employs the inverted type. With this architecture, a higher boost voltage gain is able to be achieved. A derivation of the operating principles of the converter, analyses of its topology, as well as a closed-loop control designs are performed in this study. Furthermore, simulations and experiments are also performed using input voltage of 12 V for a 120Wcircuit. A reasonable duty cycle is selected to reach output voltage of 60 V, which corresponds to static voltage gain of five. The converter achieves a maximum measured conversion efficiency of 98.7% and the full load efficiency of 89.1%

The Hard-Constraint PINNs for Interface Optimal Control Problems

We show that the physics-informed neural networks (PINNs), in combination with some recently developed discontinuity capturing neural networks, can be applied to solve optimal control problems subject to partial differential equations (PDEs) with interfaces and some control constraints. The resulting algorithm is mesh-free and scalable to different PDEs, and it ensures the control constraints rigorously. Since the boundary and interface conditions, as well as the PDEs, are all treated as soft constraints by lumping them into a weighted loss function, it is necessary to learn them simultaneously and there is no guarantee that the boundary and interface conditions can be satisfied exactly. This immediately causes difficulties in tuning the weights in the corresponding loss function and training the neural networks. To tackle these difficulties and guarantee the numerical accuracy, we propose to impose the boundary and interface conditions as hard constraints in PINNs by developing a novel neural network architecture. The resulting hard-constraint PINNs approach guarantees that both the boundary and interface conditions can be satisfied exactly and they are decoupled from the learning of the PDEs. Its efficiency is promisingly validated by some elliptic and parabolic interface optimal control problems

A cusp-capturing PINN for elliptic interface problems

In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty

An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard finite difference method is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method

A shallow physics-informed neural network for solving partial differential equations on surfaces

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We perform a series of performance study for the present methodology by solving Laplace-Beltrami equation and surface diffusion equation on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. Then we extend the present methodology to solve the advection-diffusion equation on an evolving surface with given velocity. To track the surface, we additionally introduce a prescribed hidden layer to enforce the topological structure of the surface and use the network to learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate the surfactant transport on the droplet surface under shear flow and obtain some physically plausible results

Retraction and Generalized Extension of Computing with Words

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.Comment: 13 double column pages; 3 figures; to be published in the IEEE Transactions on Fuzzy System