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Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.

 

In this paper, we investigate the problem of decoder error propagation for spatially coupled low-density parity-check (SC-LDPC) codes with sliding window decoding (SWD). This problem typically manifests itself at signal-to-noise ratios (SNRs) close to capacity under low-latency operating conditions. In this case, infrequent but severe decoder error propagation can sometimes occur. To help understand the error propagation problem in SWD of SC-LDPC codes, a multi-state Markov model is developed to describe decoder behavior and to analyze the error performance of spatially coupled LDPC codes under these conditions. We then present two approaches -check node (CN) doping and variable node (VN) doping -to combating decoder error propagation and improving decoder performance. Next we describe how the performance can be further improved by employing an adaptive approach that depends on the availability of a noiseless binary feedback channel. To illustrate the effectiveness of the doping techniques, we analyze the error performance of CN doping and VN doping using the multi-state decoder model. We then present computer simulation results showing that CN and VN doping significantly improve the performance in the operating range of interest at a cost of a small rate loss and that adaptive doping further improves the performance. We also show that the rate loss is always less than that resulting from encoder termination and can be further reduced by doping only a fraction of the VNs at each doping position in the code graph with only a minor impact on performance. Finally, we show how the encoding problem for VN doping can be greatly simplified by doping only systematic bits, with little or no performance loss. 

The seismic quality factor (Q) of the Earth’s mantle is of great importance for the understanding of the physical and chemical properties that control mantle anelasticity. The radial structure of the Earth’s Q is less well resolved compared to its wave speed structure, and large discrepancies exist among global 1-D Q models. In this study, we build a global data set of amplitude measurements of S, SS, SSS and SSSS waves using earthquakes that occurred between 2009 and 2017 with moment magnitudes ranging from 6.5 to 8.0. Synthetic seismograms for those events are computed in a 1-D reference model PREM, and amplitude ratios between observed and synthetic seismograms are calculated in the frequency domain by spectra division, with measurement windows determined based on visual inspection of seismograms. We simulate wave propagation in a global velocity model S40RTS based on SPECFEM3D and show that the average amplitude ratio as a function of epicentral distance is not sensitive to 3-D focusing and defocusing for the source–receiver configuration of the data set. This data set includes about 5500 S and SS measurements that are not affected by mantle transition zone triplications (multiple ray paths), and those measurements are applied in linear inversions to obtain a preliminary 1-D Q model QMSI. This model reveals a high Q region in the uppermost lower mantle. While model QMSI improves the overall datafit of the entire data set, it does not fully explain SS amplitudes at short epicentral distances or the amplitudes of the SSS and SSSS waves. Using forward modelling, we modify the 1-D model QMSI iteratively to reduce the overall amplitude misfit of the entire data set. The final Q model QMSF requires a stronger and thicker high Q region at depths between 600 and 900 km. This anelastic structure indicates possible viscosity layering in the mid mantle.

 

In this paper, we examine variable node (VN) doping to mitigate the error propagation problem in sliding window decoding (SWD) of spatially coupled LDPC (SC-LDPC) codes from the point of view of the encoding process. More specifically, in order to simplify the process of generating an encoded sequence with some number of doped code bits, we propose to employ systematic encoding and to limit doping to systematic bits only. Numerical results show that doping of systematic bits only achieves comparable performance to employing general (nonsystematic) encoding and full doping of all the code bits at each doping position, while benefiting from a much simpler encoding process. We then show that the inherent rate loss due to doping can be reduced by doping only a fraction of the variable nodes at each doping position with only a minor impact on performance. 

In this paper, we introduce two new methods of mitigating decoder error propagation for low-latency sliding window decoding (SWD) of spatially coupled low-density parity-check (SC-LDPC) codes. Building on the recently introduced idea of check node (CN) doping of regular SC-LDPC codes, here we employ variable node (VN) doping to fix (set to a known value) a subset of variable nodes in the coupling chain. Both of these doping methods have the effect of allowing SWD to recover from error propagation, at a cost of a slight rate loss. Experimental results show that, similar to CN doping, VN doping improves performance by up to two orders of magnitude compared to un-doped SC-LDPC codes in the typical signal-to-noise ratio operating range. Further, compared to CN doping, VN doping has the advantage of not requiring any changes to the decoding process. In addition, a log-likelihood-ratio based window extension algorithm is proposed to reduce the effect of error propagation. Using this approach, we show that decoding latency can be reduced by up to a significant fraction without suffering any loss in performance.