## 4 Interdisciplinary Applications
### Earth and Environmental Science  
**Climate Modeling (MC-PINN).** The Multiscale Climate PINN (MC-PINN) employs adaptive time stepping (\(\Delta t=30\) days \(\to 1\) h) and hierarchical training to resolve ENSO oscillations, mesoscale vortices, and extreme weather events. Compared to CMIP6 models, MC-PINN reduces equatorial Pacific sea surface temperature anomaly errors by 37% while consuming 82% less computational resources [39].  
**Earthquake Early Warning (SeisPINN).** SeisPINN decomposes seismic wavefields and inverts source parameters within 5 s of P-wave arrival, achieving 0.5 km localization accuracy during the 2025 Luding earthquake (M6.8). Traditional methods exhibit 2.1 km errors under similar conditions, highlighting PINNs' advantage in real-time geophysical inverse problems [20].  
The benchmarking results in Table 4 demonstrate PINNs' unique capability to balance physical accuracy with computational efficiency across diverse domains. In turbulence modeling, TFE-PINN achieves 62% error reduction against LES methods while operating at 19% of the computational cost (14 h vs. 72 h), attributable to its hybrid architecture combining wavelet-based multiscale decomposition with neural eddy-viscosity prediction. This performance gain intensifies in time-critical applications: SeisPINN reduces earthquake localization latency by 83% (5 s vs. 30 s) with 76% higher accuracy, enabled by real-time wavefield separation physics.  
Notably, the magnitude of improvement correlates with the intrinsic physical constraints of each domain. MRI reconstruction (ReconPINN) shows moderate SSIM gains (+0.18) due to the fundamental noise limitations of 7T scanners, whereas climate model  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Application** & **Method** & **Error Reduction** & **Compute Time** & **Reference** \\ \hline Turbulence Modeling & TFE-PINN vs. LES & 62\% & 14 h vs. 72 h & [30] \\ MRI Reconstruction & ReconPINN vs. CS & SSIM + 0.18 & 2 min vs. 15 min & [15] \\ Climate Prediction & MC-PINN vs. CMIP6 & 37\% & 18 h vs. 100 h & [39] \\ Seismic Localization & SeisPINN vs. Traditional & 76\% & 5 s vs. 30 s & [20] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Cross-Domain Performance Benchmarking of PINN Applications.  
ing (MC-PINN) achieves 37% error reduction through adaptive time-step integration that traditional CMIP6 models lack. These variations highlight context-dependent optimization strategies--turbulence simulations prioritize spatial resolution, while seismic systems demand temporal precision.  
However, two critical patterns emerge: (1) The compute time reductions (5-8\(\times\)) consistently surpass error improvements (1.5-3\(\times\)), suggesting PINNs excel more in accelerating simulations than enhancing absolute accuracy; (2) Performance scalability depends on the alignment between neural architectures and domain-specific physics, as seen in TFE-PINN's explicit turbulence closure modeling versus MC-PINN's implicit hierarchical learning. Open challenges persist in quantifying uncertainty for mission-critical deployments, particularly where PINN speed-accuracy tradeoffs intersect with safety thresholds (e.g., hypersonic thermal protection).