Md Ashiqur Rahman

Md Ashiqur Rahman, Manuel A. Florez, Anima Anandkumar et al.

We propose the generative adversarial neural operator (GANO), a generative model paradigm for learning probabilities on infinite-dimensional function spaces. The natural sciences and engineering are known to have many types of data that are sampled from infinite-dimensional function spaces, where classical finite-dimensional deep generative adversarial networks (GANs) may not be directly applicable. GANO generalizes the GAN framework and allows for the sampling of functions by learning push-forward operator maps in infinite-dimensional spaces. GANO consists of two main components, a generator neural operator and a discriminator neural functional. The inputs to the generator are samples of functions from a user-specified probability measure, e.g., Gaussian random field (GRF), and the generator outputs are synthetic data functions. The input to the discriminator is either a real or synthetic data function. In this work, we instantiate GANO using the Wasserstein criterion and show how the Wasserstein loss can be computed in infinite-dimensional spaces. We empirically study GANO in controlled cases where both input and output functions are samples from GRFs and compare its performance to the finite-dimensional counterpart GAN. We empirically study the efficacy of GANO on real-world function data of volcanic activities and show its superior performance over GAN.

Md Ashiqur Rahman, Zachary E. Ross, Kamyar Azizzadenesheli

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data efficiency, and robustness with respect to hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in an average of 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of the art. On Navier-Stokes 3D spatiotemporal operator learning task, we show U-NO provides 37% improvement over the state of art methods.

Md Ashiqur Rahman, Jasorsi Ghosh, Hrishikesh Viswanath et al.

We address the problem of generating 3D human motions in dyadic activities. In contrast to the concurrent works, which mainly focus on generating the motion of a single actor from the textual description, we generate the motion of one of the actors from the motion of the other participating actor in the action. This is a particularly challenging, under-explored problem, that requires learning intricate relationships between the motion of two actors participating in an action and also identifying the action from the motion of one actor. To address these, we propose partner conditioned motion operator (PaCMO), a neural operator-based generative model which learns the distribution of human motion conditioned by the partner's motion in function spaces through adversarial training. Our model can handle long unlabeled action sequences at arbitrary time resolution. We also introduce the "Functional Frechet Inception Distance" ($F^2ID$) metric for capturing similarity between real and generated data for function spaces. We test PaCMO on NTU RGB+D and DuetDance datasets and our model produces realistic results evidenced by the $F^2ID$ score and the conducted user study.

Hrishikesh Viswanath, Md Ashiqur Rahman, Abhijeet Vyas et al.

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach (~1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

Md Ashiqur Rahman, Raymond A. Yeh

In computer vision, models must be able to adapt to changes in image resolution to effectively carry out tasks such as image segmentation; This is known as scale-equivariance. Recent works have made progress in developing scale-equivariant convolutional neural networks, e.g., through weight-sharing and kernel resizing. However, these networks are not truly scale-equivariant in practice. Specifically, they do not consider anti-aliasing as they formulate the down-scaling operation in the continuous domain. To address this shortcoming, we directly formulate down-scaling in the discrete domain with consideration of anti-aliasing. We then propose a novel architecture based on Fourier layers to achieve truly scale-equivariant deep nets, i.e., absolute zero equivariance-error. Following prior works, we test this model on MNIST-scale and STL-10 datasets. Our proposed model achieves competitive classification performance while maintaining zero equivariance-error.

Md Ashiqur Rahman, Chiao-An Yang, Michael N. Cheng et al.

Scale variation is a fundamental challenge in computer vision. Objects of the same class can have different sizes, and their perceived size is further affected by the distance from the camera. These variations are local to the objects, i.e., different object sizes may change differently within the same image. To effectively handle scale variations, we present a deep equilibrium canonicalizer (DEC) to improve the local scale equivariance of a model. DEC can be easily incorporated into existing network architectures and can be adapted to a pre-trained model. Notably, we show that on the competitive ImageNet benchmark, DEC improves both model performance and local scale consistency across four popular pre-trained deep-nets, e.g., ViT, DeiT, Swin, and BEiT. Our code is available at https://github.com/ashiq24/local-scale-equivariance.

Md Ashiqur Rahman, Raymond A. Yeh

Downsampling layers are crucial building blocks in CNN architectures, which help to increase the receptive field for learning high-level features and reduce the amount of memory/computation in the model. In this work, we study the generalization of the uniform downsampling layer for group equivariant architectures, e.g., G-CNNs. That is, we aim to downsample signals (feature maps) on general finite groups with anti-aliasing. This involves the following: (a) Given a finite group and a downsampling rate, we present an algorithm to form a suitable choice of subgroup. (b) Given a group and a subgroup, we study the notion of bandlimited-ness and propose how to perform anti-aliasing. Notably, our method generalizes the notion of downsampling based on classical sampling theory. When the signal is on a cyclic group, i.e., periodic, our method recovers the standard downsampling of an ideal low-pass filter followed by a subsampling operation. Finally, we conducted experiments on image classification tasks demonstrating that the proposed downsampling operation improves accuracy, better preserves equivariance, and reduces model size when incorporated into G-equivariant networks

Md Ashiqur Rahman, Robert Joseph George, Mogab Elleithy et al.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-Bénard convection, we found CoDA-NO to outperform existing methods by over 36%.

Akm Ashiquzzaman, Abdul Kawsar Tushar, Md Ashiqur Rahman

Handwritten character recognition has been the center of research and a benchmark problem in the sector of pattern recognition and artificial intelligence, and it continues to be a challenging research topic. Due to its enormous application many works have been done in this field focusing on different languages. Arabic, being a diversified language has a huge scope of research with potential challenges. A convolutional neural network model for recognizing handwritten numerals in Arabic language is proposed in this paper, where the dataset is subject to various augmentation in order to add robustness needed for deep learning approach. The proposed method is empowered by the presence of dropout regularization to do away with the problem of data overfitting. Moreover, suitable change is introduced in activation function to overcome the problem of vanishing gradient. With these modifications, the proposed system achieves an accuracy of 99.4\% which performs better than every previous work on the dataset.