As distributed optimization scales to meet the demands of Large Language Model (LLM) training, hardware failures become increasingly non-negligible. Existing fault-tolerant training methods often introduce significant computational or memory overhead, demanding additional resources. To address this challenge, we propose **Me**mory- and **C**omputation- **e**fficient **F**ault-tolerant **O**ptimization (**MeCeFO**), a novel algorithm that ensures robust training with minimal overhead. When a computing node fails, MeCeFO seamlessly transfers its training task to a neighboring node while employing memory- and computation-efficient algorithmic optimizations to minimize the extra workload imposed on the neighboring node handling both tasks. MeCeFO leverages three key algorithmic designs: (i) Skip-connection, which drops the multi-head attention (MHA) module during backpropagation for memory- and computation-efficient approximation; (ii) Recomputation, which reduces activation memory in feedforward networks (FFNs); and (iii) Low-rank gradient approximation, enabling efficient estimation of FFN weight matrix gradients. Theoretically, MeCeFO matches the convergence rate of conventional distributed training, with a rate of $\mathcal{O}(1/\sqrt{nT})$, where $n$ is the data parallelism size and $T$ is the number of iterations. Empirically, MeCeFO maintains robust performance under high failure rates, incurring only a 4.18\% drop in throughput, demonstrating $5.0\times$ to $6.7\times$ greater resilience than previous SOTA approaches.

The normalized difference vegetation index (NDVI) serves as an essential measure for vegetation assessment and plays a pivotal role in environmental monitoring, ecosystem conservation, and the advancement of sustainable practices. However, NDVI forecasting typically presents a spatio-temporal challenge. While traditional techniques, such as convolutional long short-term memory (LSTM) and graph neural networks (GNNs) are frequently utilized to address this issue, the explicit spatial attributes inherent in the geographic coordinates of observation sites are often neglected in existing studies. To fill this research gap, we introduce an innovative time-aware model, fuzzy convolutional neural network long short-term memory (CNN-LSTM), designed specifically for NDVI prediction within the Chinese context. This model leverages the adaptive neuro-fuzzy inference system (ANFIS) to encapsulate the spatial nuances presented by the latitude, longitude, and elevation of the observation points. It also harnesses the power of 1-D convolution and LSTM to delineate temporal patterns. We incorporate a gate control mechanism to effectively blend the spatial intelligence rendered by ANFIS with the temporal insights captured by CNN-LSTM. We also combine deep neural fuzzy systems with traditional temporal neural networks. A comparative analysis spanning several temporal intervals highlights the superior performance of our proposed model in spatio-temporal forecasting in relation to conventional methods. Subsequent empirical evaluations confirm that the model has strong generalizability across diverse provinces.