Neural networks have emerged as powerful surrogates for solving partial differential equations (PDEs), offering significant computational speedups over traditional methods. However, these models suffer from a critical limitation: error accumulation during long-term rollouts, where small inaccuracies compound exponentially, eventually causing complete divergence from physically valid solutions. We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step by formulating correction as a linearized inverse problem based on PDE residuals. Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase, reducing computational overhead by two orders of magnitude compared to standard correction approaches. Across three representative PDE systems, including Navier-Stokes fluid dynamics, wave equations, and the chaotic Kuramoto-Sivashinsky equation, PhysicsCorrect reduces prediction errors by up to 100× while adding negligible inference time (under 5%). The framework integrates seamlessly with diverse architectures, including Fourier Neural Operators, UNets, and Vision Transformers, effectively transforming unstable neural surrogates into reliable simulation tools that bridge the gap between deep learning's computational efficiency and the physical fidelity demanded by practical scientific applications.

Solutions of wave equations (e.g., wavefields) are invaluable to imaging and inverting the subsurface. The main challenge in attaining such solutions is the exponential increase in computational cost with finer discretization. Being discretization invariant, the recently developed physics-informed neural networks (PINNs) framework offers accurate and more flexible PDE solutions than conventional solvers. However, they are challenged by the relatively slow convergence and the need to perform additional training for other PDE parameters (velocity models). To address this limitation, we introduce a PINN framework that utilizes latent representations of the PDE parameters (velocity models) as additional inputs into the PINN model and performs training over a distribution of viable velocity models. We use a two-stage training scheme in which, we first learn a latent representation for a distribution of velocity models. Then, we train a physics-informed neural network over inputs given by randomly drawn samples from the coordinate space within the solution domain and samples from the learned latent representation of the velocity models. Through numerical tests and benchmarking against several existing algorithms, we demonstrate that the proposed framework provides up to three times the speed up and an order of magnitude accuracy improvement. The proposed framework retains the flexibility and accuracy features of the functional representation of PINN solutions while gaining a generalization feature to adapt to various velocity models efficiently.

Microseismic source imaging plays a significant role in passive seismic monitoring. However, such a process is prone to failure due to aliasing when dealing with sparsely measured data. Thus, we propose a direct microseismic imaging framework based on physics-informed neural networks (PINNs), which can generate focused source images, even with very sparse recordings. We use the PINNs to represent a multi-frequency wavefield and then apply inverse Fourier transform to extract the source image. To be more specific, we modify the representation of the frequency-domain wavefield to inherently satisfy the boundary conditions (the measured data on the surface) by means of a hard constraint, which helps to avoid the difficulty in balancing the data and PDE losses in PINNs. Furthermore, we propose the causality loss implementation with respect to depth to enhance the convergence of PINNs. The numerical experiments on the Overthrust model show that the method can admit reliable and accurate source imaging for single- or multiple- sources and even in passive monitoring settings. Compared with the time-reversal method, the results of the proposed method are consistent with numerical methods but less noisy. Then, we further apply our method to hydraulic fracturing monitoring field data, and demonstrate that our method can correctly image the source with fewer artifacts.

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional $10\sim100\times$ speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

The computation of the seismic wavefield by solving the Helmholtz equation is crucial to many practical applications, e.g., full waveform inversion. Physics-informed neural networks (PINNs) provide functional wavefield solutions represented by neural networks (NNs), but their convergence is slow. To address this problem, we propose a modified PINN using multiplicative filtered networks, which embeds some of the known characteristics of the wavefield in training, e.g., frequency, to achieve much faster convergence. Specifically, we use the Gabor basis function due to its proven ability to represent wavefields accurately and refer to the implementation as GaborPINN. Meanwhile, we incorporate prior information on the frequency of the wavefield into the design of the method to mitigate the influence of the discontinuity of the represented wavefield by GaborPINN. The proposed method achieves up to a two-magnitude increase in the speed of convergence as compared with conventional PINNs.

Full waveform inversion (FWI) has the potential to provide high-resolution subsurface model estimations. However, due to limitations in observation, e.g., regional noise, limited aperture, and band-limited data, it is hard to obtain the desired high-resolution model with FWI. To address this challenge, we propose a new paradigm for FWI regularized by generative diffusion model. Specifically, we pre-train a diffusion model in a fully unsupervised manner on a prior velocity model distribution that represents our expectations of the subsurface and then adapt it to the seismic observations by incorporating the FWI into the sampling process of the generative diffusion models. What makes diffusion models uniquely appropriate for such an implementation is that the generative process retains the form and dimensions of the velocity model. Numerical examples demonstrate that our method can outperform the conventional FWI with only negligible additional computational cost. Even in cases of very sparse observations or observations with strong noise, the proposed method could still reconstruct a high-quality subsurface model. Thus, we can incorporate our prior expectations of the solutions in an efficient manner. We further test this approach on field data, which demonstrates the effectiveness of the proposed method.

Imaging is a crucial inversion-based task in fields ranging from medical to structure investigations, and Earth discovery. The exploding reflector assumption provides a direct imaging approach for zero-offset (coincident source-receiver) data, like ground penetrating radar (GPR) data. In imaging, however, we face aliasing problems when the data are coarsely sampled. Formulating the corresponding frequency-domain wavefield as a neural network (NN) function of the lateral and depth coordinates, as well as frequency, allows us to use the physics-informed neural network (PINN) framework to obtain images of the subsurface. In this case, we use a modified Helmholtz equation that incorporates the data on the Earth surface (hard constraint) as the loss function to optimize the NN function. This modified Helmholtz formulation allows us to avoid the inherent weaknesses that PINN has in handling boundary conditions, like the data on the surface as an additional loss term. The frequency dimension allows for image reconstruction by directly summing the wavefield over frequencies (the zero-time imaging condition). This zero-offset implementation serves as a proof of concept for later extensions to prestack data using the double square-root equation.

Physics-informed neural networks (PINNs) can offer approximate multidimensional functional solutions to the Helmholtz equation that are flexible, require low memory, and have no limitations on the shape of the solution space. However, the neural network (NN) training can be costly and the cost dramatically increases as we train for multi-frequency wavefields by adding frequency as an additional input to the NN multi-dimensional function. In this case, the often large variation of the wavefield features (specifically wavelength) with frequency adds more complexity to the NN training. Thus, we propose a new loss function for the NN multidimensional input training that allows us to seamlessly include frequency as a dimension. We specifically utilize the linear relation between frequency and wavenumber (the wavefield space representation) to incorporate a reference frequency scaling to the loss function. As a result, the effective wavenumber of the wavefield solution as a function of frequency remains almost stationary, which reduces the learning burden on the NN function. We demonstrate the effectiveness of this modified loss function on a layered model.

Considering the high-density and high-dynamic feature of cooperative unmanned aerial vehicles (UAVs) swarm, also referred to as flocking of flying ad hoc networks (FANETs), reliable medium access control (MAC) protocol design for network connectivity maintaining and network information sharing is a challenging issue. In this article, we propose a self-adaptive carrier sense multiple access with collision avoidance (CSMA/CA)-based MAC protocol for flocking of FANET, namely, FMAC, to provide reliable broadcast information service under density-varying flocking scenarios. To represent the varying trend of UAV density during flocking, we define the collective neighboring potential (CNP) in the FMAC protocol. Specifically, at the beginning of each period, each UAV computes the current CNP based on available neighbors' motion states. Then, the value of CNP at the start of the next period regarding the same neighbors is predicted using UAV's kinetic equation. After that, each UAV can update the contention window (CW) size by comparing the current CNP and the predicted CNP, and CW will be decreased (increased) if the current CNP is larger (smaller) than the predicted one for enough period. The simulation results show that the proposed FMAC protocol can ensure high successful transmission probability under density-varying flocking scenarios and outperforms the typical MAC solutions.

Spiking neural P systems with astrocytes (SNPA systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. In this work, we investigate the reversibility in SNPA systems as w