Khalid Rafiq

Khalid Rafiq, Aditya G. Nair

Latent diffusion and flow matching have emerged as leading approaches for synthetic turbulence generation, yet they systematically under-represent dissipation-range amplitudes. We introduce a latent flow matching framework with a spectrally regularized compression stage that directly targets this failure mode. On a 256^2 DNS dataset at Re_f \approx 2250, replacing an MSE-trained VAE with a zone-weighted log-spectral objective raises deep-dissipation retained spectral power from 25% to 94% in reconstruction and from 20% to 79% in unconditional generation. The improved latent representation also yields a substantially better sampling cost-fidelity tradeoff: the MSE-trained latent space imposes a fundamental quality ceiling near DD bias -0.70 that no integrator or step-count can overcome, while the spectrally regularized latent space reaches DD bias -0.117 at just 20 function evaluations. Mechanistically, encoder-decoder swap experiments show that the improvement is driven primarily by encoder-induced latent reorganization rather than decoder capacity, while a support-amplitude decomposition reveals that MSE-trained models behave as conservative suppression models, minimizing pointwise error by attenuating intermittent high-wavenumber structure. Both pipelines recover the second-order structure function and the correct sign of S_3, indicating the correct cascade direction without explicit supervision. A small residual gap in the magnitude of S_3 suggests that phase-coherent triadic organization remains a complementary axis to amplitude fidelity for future generative turbulence models.

Khalid Rafiq, Wenjing Liao, Aditya G. Nair

We introduce Flexi-VAE, a data-driven framework for efficient single-shot forecasting of nonlinear parametric partial differential equations (PDEs), eliminating the need for iterative time-stepping while maintaining high accuracy and stability. Flexi-VAE incorporates a neural propagator that advances latent representations forward in time, aligning latent evolution with physical state reconstruction in a variational autoencoder setting. We evaluate two propagation strategies, the Direct Concatenation Propagator (DCP) and the Positional Encoding Propagator (PEP), and demonstrate, through representation-theoretic analysis, that DCP offers superior long-term generalization by fostering disentangled and physically meaningful latent spaces. Geometric diagnostics, including Jacobian spectral analysis, reveal that propagated latent states reside in regions of lower decoder sensitivity and more stable local geometry than those derived via direct encoding, enhancing robustness for long-horizon predictions. We validate Flexi-VAE on canonical PDE benchmarks, the 1D viscous Burgers equation and the 2D advection-diffusion equation, achieving accurate forecasts across wide parametric ranges. The model delivers over 50x CPU and 90x GPU speedups compared to autoencoder-LSTM baselines for large temporal shifts. These results position Flexi-VAE as a scalable and interpretable surrogate modeling tool for accelerating high-fidelity simulations in computational fluid dynamics (CFD) and other parametric PDE-driven applications, with extensibility to higher-dimensional and more complex systems.