Parametric differential equations of the form $\frac{\partial u}{\partial t} = f(u, x, t, p)$ are fundamental in science and engineering. While deep learning frameworks like the Fourier Neural Operator (FNO) efficiently approximate differential equation solutions, they struggle with inverse problems, sensitivity calculations $\frac{\partial u}{\partial p}$, and concept drift. We address these challenges by introducing a novel sensitivity loss regularizer, demonstrated through Sensitivity-Constrained Fourier Neural Operators (SC-FNO). Our approach maintains high accuracy for solution paths and outperforms both standard FNO and FNO with Physics-Informed Neural Network regularization. SC-FNO exhibits superior performance in parameter inversion tasks, accommodates more complex parameter spaces (tested with up to 82 parameters), reduces training data requirements, and decreases training time while maintaining accuracy. These improvements apply across various differential equations and neural operators, enhancing their reliability without significant computational overhead (30%–130% extra training time per epoch). Models and selected experiment code are available at: [https://github.com/AMBehroozi/SC_Neural_Operators](https://github.com/AMBehroozi/SC_Neural_Operators).

Downward longwave radiation (DLR) at the surface is a key variable of interest in fields, such as hydrology and climate research. However, existing DLR estimation methods and DLR products are still problematic in terms of both accuracy and spatiotemporal resolution. In this article, we propose a deep convolutional neural network (DCNN)-based method to estimate hourly DLR at 5-km spatial resolution from top of atmosphere (TOA) brightness temperature (BT) of the Himawari-8/Advanced Himawari Imager (AHI) thermal channels, combined with near-surface air temperature and dew point temperature of ERA5 and elevation data. Validation results show that the DCNN-based method outperforms popular random forest and multilayer perceptron-based methods and that our proposed scheme integrating multisource data outperforms that only using remote sensing TOA observations or surface meteorological data. Compared with state-of-the-art CERES-SYN and ERA5-land DLR products, the estimated DLR by our proposed DCNN-based method with physical multisource inputs has higher spatiotemporal resolution and accuracy, with correlation coefficient (CC) of 0.95, root-mean-square error (RMSE) of 17.2 W/m2, and mean bias error (MBE) of −0.8 W/m2in the testing period on the Tibetan Plateau.

The Soil Moisture Active Passive (SMAP) mission measures important soil moisture data globally. SMAP's products might not always perform better than land surface models (LSM) when evaluated against in situ measurements. However, we hypothesize that SMAP presents added value for long-term soil moisture estimation in a data fusion setting as evaluated by in situ data. Here, with the help of a time series deep learning (DL) method, we created a seamlessly extended SMAP data set to test this hypothesis and, importantly, gauge whether such benefits extend to years beyond SMAP's limited lifespan. We first show that the DL model, called long short-term memory (LSTM), can extrapolate SMAP for several years and the results are similar to the training period. We obtained prolongation results with low-performance degradation where SMAP itself matches well with in situ data. Interannual trends of root-zone soil moisture are surprisingly well captured by LSTM. In some cases, LSTM's performance is limited by SMAP, whose main issue appears to be its shallow sensing depth. Despite this limitation, a simple average between LSTM and an LSM Noah frequently outperforms Noah alone. Moreover, Noah combined with LSTM is more skillful than when it is combined with another LSM. Over sparsely instrumented sites, the Noah-LSTM combination shows a stronger edge. Our results verified the value of LSTM-extended SMAP data. Moreover, DL is completely data driven and does not require structural assumptions. As such, it has its unique potential for long-term projections and may be applied synergistically with other model-data integration techniques.