Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, each solved on the same mesh as its CFD reference, the FVM-loss model reproduces the reference flow and temperature fields with mean field-averaged normalized root-mean-square errors (nRMSEs) near 3%, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model attains lower rollout errors than the supervised baseline in almost every field, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.
The sustainability of the Web has drawn growing attention, as large language model (LLM)-based web services leave a substantial carbon footprint. These services typically run in geo-distributed data centers whose grid carbon intensity varies across regions and over time, so routing each query to a cleaner region or a smaller model can reduce emissions. A conversation, however, is a stateful web session: each turn reuses the key--value (KV) cache of its history, which resides only where earlier turns were served. Serving a turn on another deployment requires recomputing this history, whose carbon cost grows with the conversation and can outweigh the saving of the move. Existing carbon-aware LLM routers treat a conversation's queries as independent and ignore this cost. We present CarbonRouter, an online framework that jointly selects the model, GPU type, and region for each query to minimize carbon emissions. CarbonRouter introduces a cache-aware carbon cost model, which charges history recomputation to every candidate deployment using cached-token counts from the serving backends, and uses a lightweight gate that keeps a session on its cached deployment when it would likely move back at the next turn and rebuild its history again. We evaluate CarbonRouter in trace-driven experiments on two real-world conversation datasets with five LLMs, three GPU types, and four regions, where it achieves the lowest carbon of all policies. CarbonRouter reduces CO$_2$ by 16.1\% on average relative to FairCarbon, a state-of-the-art carbon-aware router with the same action space, while also improving predicted answer quality and estimated latency. Moreover, the saving rises further to 23.1\% for sessions longer than 30 turns.