Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.

It is widely recognized that as electronic systems’ operating frequency shifts to microwave and millimeter wave bands, the integration of ferrite passive devices with semiconductor solid state active devices holds significant advantages in improved miniaturization, bandwidth, speed, power and production costs, among others. Traditionally, ferrites have been employed in discrete bulk form, despite attempts to integrate ferrite as films within microwave integrated circuits. Technical barriers remain centric to the incompatibility between ferrite and semiconductor materials and their processing protocols. In this review, we present past and present efforts at ferrite integration with semiconductor platforms with the aim to identify the most promising paths to realizing the complete integration of on-chip ferrite and semiconductor devices, assemblies and systems.

Copper ferrite films have been deposited on ͑100͒ MgO substrates by pulsed-laser deposition. The oxygen pressure used in deposition was varied from 1 to 120 mTorr with the substrate temperature fixed at 700°C. Magnetization values are measured to increase with oxygen pressure, reaching a maximum value of 2480 G, which is a 42% increase over the bulk equilibrium value. Extended x-ray absorption spectroscopy shows that the Cu cation inversion ␦ ͓defined as ͑Cu 1−␦ Fe ␦ ͒ tet ͓Cu ␦ Fe 2−␦ ͔ oct O 4 ͔ decreases monotonically from 0.72 to 0.55 with increasing saturation magnetization.

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