Incorporating second-order curvature information into machine learning optimization algorithms can be subtle, and doing so naïvely can lead to high per-iteration costs associated with forming the Hessian and performing the associated linear system solve. To address this, we introduce ADAHESSIAN, a new stochastic optimization algorithm. ADAHESSIAN directly incorporates approximate curvature information from the loss function, and it includes several novel performance-improving features, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a spatial averaging to reduce the variance of the second derivative; and (iii) a root-mean-square exponential moving average to smooth out variations of the second-derivative across different iterations. We perform extensive tests on NLP, CV, and recommendation system tasks, and ADAHESSIAN achieves state-of-the-art results. In particular, we find that ADAHESSIAN: (i) outperforms AdamW for transformers by0.13/0.33 BLEU score on IWSLT14/WMT14, 2.7/1.0 PPLon PTB/Wikitext-103; (ii) outperforms AdamW for Squeeze-Bert by 0.41 points on GLUE; (iii) achieves 1.45%/5.55%higher accuracy on ResNet32/ResNet18 on Cifar10/ImageNetas compared to Adam; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. The cost per iteration of ADAHESSIANis comparable to first-order methods, and ADAHESSIAN exhibits improved robustness towards variations in hyperparameter values. The code for ADAHESSIAN is open-sourced and publicly-available [1].

While deep learning has shown tremendous success in a wide range of domains, it remains a grand challenge to incorporate physical principles in a systematic manner to the design, training, and inference of such models. In this paper, we aim to predict turbulent flow by learning its highly nonlinear dynamics from spatiotemporal velocity fields of large-scale fluid flow simulations of relevance to turbulence modeling and climate modeling. We adopt a hybrid approach by marrying two well-established turbulent flow simulation techniques with deep learning. Specifically, we introduce trainable spectral filters in a coupled model of Reynolds-averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES), followed by a specialized U-net for prediction. Our approach, which we call Turbulent-Flow Net, is grounded in a principled physics model, yet offers the flexibility of learned representations. We compare our model with state-of-the-art baselines and observe significant reductions in error for predictions 60 frames ahead. Most importantly, our method predicts physical fields that obey desirable physical characteristics, such as conservation of mass, whilst faithfully emulating the turbulent kinetic energy field and spectrum, which are critical for accurate prediction of turbulent flows.

We propose MESHFREEFLOWNET, a novel deep learning-based super-resolution framework to generate continuous (grid-free) spatio-temporal solutions from the low-resolution inputs. While being computationally efficient, MESHFREEFLOWNET accurately recovers the fine-scale quantities of interest. MESHFREEFLOWNET allows for: (i) the output to be sampled at all spatio-temporal resolutions, (ii) a set of Partial Differential Equation (PDE) constraints to be imposed, and (iii) training on fixed-size inputs on arbitrarily sized spatio-temporal domains owing to its fully convolutional encoder. We empirically study the performance of MESHFREEFLOWNET on the task of super-resolution of turbulent flows in the Rayleigh-Bénard convection problem. Across a diverse set of evaluation metrics, we show that MESHFREEFLOWNET significantly outperforms existing baselines. Furthermore, we provide a large scale implementation of MESHFREEFLOWNET and show that it efficiently scales across large clusters, achieving 96.80% scaling efficiency on up to 128 GPUs and a training time of less than 4 minutes. We provide an opensource implementation of our method that supports arbitrary combinations of PDE constraints.