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The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the 2D Burgers equation, the equation of 2D channel flow over an obstacle, and the equation of 3D intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through Single and Coupled Klein-Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data.

The partial differential equation (PDE) plays a significantly important role in many fields of science and engineering. The conventional case of the derivation of PDE mainly relies on first principles and empirical observation. However, the development of machine learning technology allows us to mine potential control equations from the massive amounts of stored data in a fresh way. Although there has been considerable progress in the data-driven discovery of PDE, the extant literature mostly focuses on the improvements of discovery methods, without substantial breakthroughs in the discovery process itself, including the principles for the construction of candidates and how to incorporate physical priors. In this paper, through rigorous derivation of formulas, novel physically enhanced machining learning discovery methods for control equations: GSNN (Galileo Symbolic Neural Network) and LSNN (Lorentz Symbolic Neural Network) are firstly proposed based on Galileo invariance and Lorentz invariance respectively, setting forth guidelines for building the candidates of discovering equations. The adoption of mandatory embedding of physical constraints is fundamentally different from PINN in the form of the loss function, thus ensuring that the designed Neural Network strictly obeys the physical prior of invariance and enhancing the interpretability of the network. By comparing the results with PDE-NET in numerical experiments of Burgers equation and Sine-Gordon equation, it shows that the method presented in this study has better accuracy, parsimony, and interpretability.

Sulphurization processes in Cu(In,Ga)Se$_2$ thin-film solar cells has been intensively studied in the last decade as a viable alternative to the existing Ga-grading. The main advantage of using S grading is that by substituting Se with S we will achieve not only an upshift of the conduction-band minimum as done by employing Ga grading, but also a downshift of the valence-band maximum. Several existing studies stipulate that S is very often inserted in too high concentrations into Cu(In,Ga)Se$_2$ absorber by sulphurization resulting in a deteriorated device performance instead of the expected beneficial effect. However, we demonstrate here that the intense sulphurization process when accompanied by Ga-grading leads to improved electrical properties of the buffer/absorber heterojunction. More exactly, this double grading at the absorber surface leads to strong reduction of the p-doping and hence to a change in the band diagram. This work also proves that the intense sulphurization process is accompanied by strong structural and chemical changes, i.e., by the formation of a S-rich CuIn(S,Se)$_2$ compound at the absorber surface. Finally, all these experimental findings were complemented by ab-initio calculations of the conduction-band and valence-band offsets between absorber and buffer obtained by using density functional theory. Hence, the present work opens up new possibilities for synthesizing Cu(In,Ga)(Se,S)2 solar cells with superior cell performance when using an intense sulphurization process.

We employ physics-informed neural networks (PINNs) to simulate the incompressible flows ranging from laminar to turbulent flows. We perform PINN simulations by considering two different formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. We refer to these specific PINNs for the Navier-Stokes flow nets as NSFnets. Analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. These two different forms of the Navier-Stokes equations together with the initial and boundary conditions are embedded into the loss function of the PINNs. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without splitting the equations. We obtain good accuracy of the NSFnet simulation results upon convergence of the loss function, verifying that NSFnets can effectively simulate complex incompressible flows using either the VP or the VV formulations. We also perform a systematic study on the weights used in the loss function for the data/physics components and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. Our results suggest that the accuracy of NSFnets, for both laminar and turbulent flows, can be improved with proper tuning of weights (manual or dynamic) in the loss function.

A universal rule-based self-learning approach using deep reinforcement learning (DRL) is proposed for the first time to solve nonlinear ordinary differential equations and partial differential equations. The solver consists of a deep neural network-structured actor that outputs candidate solutions, and a critic derived only from physical rules (governing equations and boundary and initial conditions). Solutions in discretized time are treated as multiple tasks sharing the same governing equation, and the current step parameters provide an ideal initialization for the next owing to the temporal continuity of the solutions, which shows a transfer learning characteristic and indicates that the DRL solver has captured the intrinsic nature of the equation. The approach is verified through solving the Schr\"odinger, Navier-Stokes, Burgers', Van der Pol, and Lorenz equations and an equation of motion. The results indicate that the approach gives solutions with high accuracy, and the solution process promises to get faster.