Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interface

Partial differential equations (PDEs) are commonly employed to model complex industrial systems characterized by multivariable dependence. Existing physics-informed neural networks (PINNs) excel in solving PDEs in a homogeneous medium. However, their feasibility is diminished when PDE parameters are unknown due to a lack of physical attributions and time-varying interface is unavailable arising from heterogeneous media. To this end, we propose a data-physics-hybrid method, physically informed synchronic-adaptive learning (PISAL), to solve PDEs for industrial systems modeling in heterogeneous media. First, Net1, Net2, and NetI, are constructed to approximate the solutions satisfying PDEs and the interface. Net1 and Net2 are utilized to synchronously learn each solution satisfying PDEs with diverse parameters, while NetI is employed to adaptively learn the unavailable time-varying interface. Then, a criterion combined with NetI is introduced to adaptively distinguish the attributions of measurements and collocation points. Furthermore, NetI is integrated into a data-physics-hybrid loss function. Accordingly, a synchronic-adaptive learning (SAL) strategy is proposed to decompose and optimize each subdomain. Besides, we theoretically prove the approximation capability of PISAL. Extensive experimental results verify that the proposed PISAL can be used for industrial systems modeling in heterogeneous media, which faces the challenges of lack of physical attributions and unavailable time-varying interface.

A Domain-adaptive Physics-informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

Maxwell's equations are a collection of coupled partial differential equations (PDEs) that, together with the Lorentz force law, constitute the basis of classical electromagnetism and electric circuits. Effectively solving Maxwell's equations is crucial in various fields, like electromagnetic scattering and antenna design optimization. Physics-informed neural networks (PINNs) have shown powerful ability in solving PDEs. However, PINNs still struggle to solve Maxwell's equations in heterogeneous media. To this end, we propose a domain-adaptive PINN (da-PINN) to solve inverse problems of Maxwell's equations in heterogeneous media. First, we propose a location parameter of media interface to decompose the whole domain into several sub-domains. Furthermore, the electromagnetic interface conditions are incorporated into a loss function to improve the prediction performance near the interface. Then, we propose a domain-adaptive training strategy for da-PINN. Finally, the effectiveness of da-PINN is verified with two case studies.

PDE Discovery for Soft Sensors Using Coupled Physics-Informed Neural Network with Akaike's Information Criterion

Soft sensors have been extensively used to monitor key variables using easy-to-measure variables and mathematical models. Partial differential equations (PDEs) are model candidates for soft sensors in industrial processes with spatiotemporal dependence. However, gaps often exist between idealized PDEs and practical situations. Discovering proper structures of PDEs, including the differential operators and source terms, can remedy the gaps. To this end, a coupled physics-informed neural network with Akaike's criterion information (CPINN-AIC) is proposed for PDE discovery of soft sensors. First, CPINN is adopted for obtaining solutions and source terms satisfying PDEs. Then, we propose a data-physics-hybrid loss function for training CPINN, in which undetermined combinations of differential operators are involved. Consequently, AIC is used to discover the proper combination of differential operators. Finally, the artificial and practical datasets are used to verify the feasibility and effectiveness of CPINN-AIC for soft sensors. The proposed CPINN-AIC is a data-driven method to discover proper PDE structures and neural network-based solutions for soft sensors.

Evaluating robustness of support vector machines with the Lagrangian dual approach

Adversarial examples bring a considerable security threat to support vector machines (SVMs), especially those used in safety-critical applications. Thus, robustness verification is an essential issue for SVMs, which can provide provable robustness against various kinds of adversary attacks. The evaluation results obtained through the robustness verification can provide a safe guarantee for the use of SVMs. The existing verification method does not often perform well in verifying SVMs with nonlinear kernels. To this end, we propose a method to improve the verification performance for SVMs with nonlinear kernels. We first formalize the adversarial robustness evaluation of SVMs as an optimization problem. Then a lower bound of the original problem is obtained by solving the Lagrangian dual problem of the original problem. Finally, the adversarial robustness of SVMs is evaluated concerning the lower bound. We evaluate the adversarial robustness of SVMs with linear and nonlinear kernels on the MNIST and Fashion-MNIST datasets. The experimental results show that the percentage of provable robustness obtained by our method on the test set is better than that of the state-of-the-art.

Coupled Physics-informed Neural Networks for Inferring Solutions of Partial Differential Equations with Unknown Source Terms

Physics-informed neural networks (PINNs) provide a transformative development for approximating the solutions to partial differential equations (PDEs). This work proposes a coupled physics-informed neural network (C-PINN) for the nonhomogeneous PDEs with unknown dynamical source terms, which is used to describe the systems with external forces and cannot be well approximated by the existing PINNs. In our method, two neural networks, NetU and NetG, are proposed. NetU is constructed to generate a quasi-solution satisfying PDEs under study. NetG is used to regularize the training of NetU. Then, the two networks are integrated into a data-physics-hybrid cost function. Finally, we propose a hierarchical training strategy to optimize and couple the two networks. The performance of C-PINN is proved by approximating several classical PDEs.