Tanya Marwah

Tanya Marwah, Ashwini Pokle, J. Zico Kolter et al.

Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and outputs its solution. However, the architectural design space, especially given structural knowledge of the PDE family of interest, is still poorly understood. We seek to remedy this gap by studying the benefits of weight-tied neural network architectures for steady-state PDEs. To achieve this, we first demonstrate that the solution of most steady-state PDEs can be expressed as a fixed point of a non-linear operator. Motivated by this observation, we propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE as the infinite-depth fixed point of an implicit operator layer using a black-box root solver and differentiates analytically through this fixed point resulting in $\mathcal{O}(1)$ training memory. Our experiments indicate that FNO-DEQ-based architectures outperform FNO-based baselines with $4\times$ the number of parameters in predicting the solution to steady-state PDEs such as Darcy Flow and steady-state incompressible Navier-Stokes. Finally, we show FNO-DEQ is more robust when trained with datasets with more noisy observations than the FNO-based baselines, demonstrating the benefits of using appropriate inductive biases in architectural design for different neural network based PDE solvers. Further, we show a universal approximation result that demonstrates that FNO-DEQ can approximate the solution to any steady-state PDE that can be written as a fixed point equation.

Tanya Marwah, Zachary C. Lipton, Jianfeng Lu et al.

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as \emph{nonlinear elliptic variational PDEs}, whose solutions minimize an \emph{Euler-Lagrange} energy functional $\mathcal{E}(u) = \int_ΩL(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $ε$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ ε), p^{\log(1/ε)}\}}\right)$. By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, ε, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Zachary Novack, Simran Kaur, Tanya Marwah et al.

A number of competing hypotheses have been proposed to explain why small-batch Stochastic Gradient Descent (SGD)leads to improved generalization over the full-batch regime, with recent work crediting the implicit regularization of various quantities throughout training. However, to date, empirical evidence assessing the explanatory power of these hypotheses is lacking. In this paper, we conduct an extensive empirical evaluation, focusing on the ability of various theorized mechanisms to close the small-to-large batch generalization gap. Additionally, we characterize how the quantities that SGD has been claimed to (implicitly) regularize change over the course of training. By using micro-batches, i.e. disjoint smaller subsets of each mini-batch, we empirically show that explicitly penalizing the gradient norm or the Fisher Information Matrix trace, averaged over micro-batches, in the large-batch regime recovers small-batch SGD generalization, whereas Jacobian-based regularizations fail to do so. This generalization performance is shown to often be correlated with how well the regularized model's gradient norms resemble those of small-batch SGD. We additionally show that this behavior breaks down as the micro-batch size approaches the batch size. Finally, we note that in this line of inquiry, positive experimental findings on CIFAR10 are often reversed on other datasets like CIFAR100, highlighting the need to test hypotheses on a wider collection of datasets.

Ricardo Buitrago Ruiz, Tanya Marwah, Albert Gu et al.

Data-driven techniques have emerged as a promising alternative to traditional numerical methods for solving PDEs. For time-dependent PDEs, many approaches are Markovian -- the evolution of the trained system only depends on the current state, and not the past states. In this work, we investigate the benefits of using memory for modeling time-dependent PDEs: that is, when past states are explicitly used to predict the future. Motivated by the Mori-Zwanzig theory of model reduction, we theoretically exhibit examples of simple (even linear) PDEs, in which a solution that uses memory is arbitrarily better than a Markovian solution. Additionally, we introduce Memory Neural Operator (MemNO), a neural operator architecture that combines recent state space models (specifically, S4) and Fourier Neural Operators (FNOs) to effectively model memory. We empirically demonstrate that when the PDEs are supplied in low resolution or contain observation noise at train and test time, MemNO significantly outperforms the baselines without memory -- with up to 6x reduction in test error. Furthermore, we show that this benefit is particularly pronounced when the PDE solutions have significant high-frequency Fourier modes (e.g., low-viscosity fluid dynamics) and we construct a challenging benchmark dataset consisting of such PDEs.

Zhili Feng, Tanya Marwah, Nicolo Fusi et al. · harvard, microsoft-research

Modern large language models use a fixed tokenizer to effectively compress text drawn from a source domain. However, applying the same tokenizer to a new target domain often leads to inferior compression, more costly inference, and reduced semantic alignment. To address this deficiency, we introduce Sparse Sinkhorn Token Translation (S2T2). S2T2 trains a tailored tokenizer for the target domain and learns to translate between target and source tokens, enabling more effective reuse of the pre-trained next-source-token predictor. In our experiments with finetuned English language models, S2T2 improves both the perplexity and the compression of out-of-domain protein sequences, outperforming direct finetuning with either the source or target tokenizer. In addition, we find that token translations learned for smaller, less expensive models can be directly transferred to larger, more powerful models to reap the benefits of S2T2 at lower cost.

Dhruv Rohatgi, Tanya Marwah, Zachary Chase Lipton et al.

Graph neural networks (GNNs) are the dominant approach to solving machine learning problems defined over graphs. Despite much theoretical and empirical work in recent years, our understanding of finer-grained aspects of architectural design for GNNs remains impoverished. In this paper, we consider the benefits of architectures that maintain and update edge embeddings. On the theoretical front, under a suitable computational abstraction for a layer in the model, as well as memory constraints on the embeddings, we show that there are natural tasks on graphical models for which architectures leveraging edge embeddings can be much shallower. Our techniques are inspired by results on time-space tradeoffs in theoretical computer science. Empirically, we show architectures that maintain edge embeddings almost always improve on their node-based counterparts -- frequently significantly so in topologies that have ``hub'' nodes.

Tanya Marwah, Zachary C. Lipton, Andrej Risteski

Recent experiments have shown that deep networks can approximate solutions to high-dimensional PDEs, seemingly escaping the curse of dimensionality. However, questions regarding the theoretical basis for such approximations, including the required network size, remain open. In this paper, we investigate the representational power of neural networks for approximating solutions to linear elliptic PDEs with Dirichlet boundary conditions. We prove that when a PDE's coefficients are representable by small neural networks, the parameters required to approximate its solution scale polynomially with the input dimension $d$ and proportionally to the parameter counts of the coefficient networks. To this we end, we develop a proof technique that simulates gradient descent (in an appropriate Hilbert space) by growing a neural network architecture whose iterates each participate as sub-networks in their (slightly larger) successors, and converge to the solution of the PDE. We bound the size of the solution, showing a polynomial dependence on $d$ and no dependence on the volume of the domain.

Nagendra Kumar, Rakshita Nagalla, Tanya Marwah et al.

Social media is currently one of the most important means of news communication. Since people are consuming a large fraction of their daily news through social media, most of the traditional news channels are using social media to catch the attention of users. Each news channel has its own strategies to attract more users. In this paper, we analyze how the news channels use sentiment to garner users' attention in social media. We compare the sentiment of social media news posts of television, radio and print media, to show the differences in the ways these channels cover the news. We also analyze users' reactions and opinion sentiment on news posts with different sentiments. We perform our experiments on a dataset extracted from Facebook Pages of five popular news channels. Our dataset contains 0.15 million news posts and 1.13 billion users reactions. The results of our experiments show that the sentiment of user opinion has a strong correlation with the sentiment of the news post and the type of information source. Our study also illustrates the differences among the social media news channels of different types of news sources.

Gaurav Mittal, Shubham Agrawal, Anuva Agarwal et al.

Recent years have witnessed some exciting developments in the domain of generating images from scene-based text descriptions. These approaches have primarily focused on generating images from a static text description and are limited to generating images in a single pass. They are unable to generate an image interactively based on an incrementally additive text description (something that is more intuitive and similar to the way we describe an image). We propose a method to generate an image incrementally based on a sequence of graphs of scene descriptions (scene-graphs). We propose a recurrent network architecture that preserves the image content generated in previous steps and modifies the cumulative image as per the newly provided scene information. Our model utilizes Graph Convolutional Networks (GCN) to cater to variable-sized scene graphs along with Generative Adversarial image translation networks to generate realistic multi-object images without needing any intermediate supervision during training. We experiment with Coco-Stuff dataset which has multi-object images along with annotations describing the visual scene and show that our model significantly outperforms other approaches on the same dataset in generating visually consistent images for incrementally growing scene graphs.

Tanya Marwah, Gaurav Mittal, Vineeth N. Balasubramanian

This paper proposes a network architecture to perform variable length semantic video generation using captions. We adopt a new perspective towards video generation where we allow the captions to be combined with the long-term and short-term dependencies between video frames and thus generate a video in an incremental manner. Our experiments demonstrate our network architecture's ability to distinguish between objects, actions and interactions in a video and combine them to generate videos for unseen captions. The network also exhibits the capability to perform spatio-temporal style transfer when asked to generate videos for a sequence of captions. We also show that the network's ability to learn a latent representation allows it generate videos in an unsupervised manner and perform other tasks such as action recognition. (Accepted in International Conference in Computer Vision (ICCV) 2017)