Anthony Zhou

Sungwon Park, Anthony Zhou, Hongjoong Kim et al.

Neural operators have emerged as a powerful paradigm for learning discretization-invariant function-to-function mappings in scientific computing. However, many practical systems are inherently stochastic, making principled uncertainty quantification essential for reliable deployment. To address this, we introduce a simple add-on, the diffusion last layer (DLL), a lightweight probabilistic head that can be attached to arbitrary neural operator backbones to model predictive uncertainty. Motivated by the relative smoothness and low-dimensional structure often exhibited by PDE solution distributions, DLL parameterizes the conditional output distribution directly in function space through a low-rank Karhunen-Loève expansion, enabling efficient and expressive uncertainty modeling. Across stochastic PDE operator learning benchmarks, DLL improves generalization and uncertainty-aware prediction. Moreover, even in deterministic long-horizon rollout settings, DLL enhances rollout stability and provides meaningful estimates of epistemic uncertainty for backbone neural operators.

Anthony Zhou, Amir Barati Farimani · cmu

The growth of global consumption has motivated important applications of deep learning to smart manufacturing and machine health monitoring. In particular, analyzing vibration data offers great potential to extract meaningful insights into predictive maintenance by the detection of bearing faults. Deep learning can be a powerful method to predict these mechanical failures; however, they lack generalizability to new tasks or datasets and require expensive, labeled mechanical data. We address this by presenting a novel self-supervised pretraining and fine-tuning framework based on transformer models. In particular, we investigate different tokenization and data augmentation strategies to reach state-of-the-art accuracies using transformer models. Furthermore, we demonstrate self-supervised masked pretraining for vibration signals and its application to low-data regimes, task adaptation, and dataset adaptation. Pretraining is able to improve performance on scarce, unseen training samples, as well as when fine-tuning on fault classes outside of the pretraining distribution. Furthermore, pretrained transformers are shown to be able to generalize to a different dataset in a few-shot manner. This introduces a new paradigm where models can be pretrained on unlabeled data from different bearings, faults, and machinery and quickly deployed to new, data-scarce applications to suit specific manufacturing needs.

Zijie Li, Anthony Zhou, Amir Barati Farimani · cmu

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion probabilistic model has been shown to enhance the temporal stability of neural solvers, while its stochastic inference mechanism enables ensemble predictions and uncertainty quantification. In principle, such training involves sampling a series of discretized diffusion timesteps during both training and inference, inevitably increasing computational overhead. In addition, most diffusion models apply isotropic Gaussian noise on structured, uniform grids, limiting their adaptability to irregular domains. We propose a latent diffusion model for PDE simulation that embeds the PDE state in a lower-dimensional latent space, which significantly reduces computational costs. Our framework uses an autoencoder to map different types of meshes onto a unified structured latent grid, capturing complex geometries. By analyzing common diffusion paths, we propose to use a coarsely sampled noise schedule from flow matching for both training and testing. Numerical experiments show that the proposed model outperforms several deterministic baselines in both accuracy and long-term stability, highlighting the potential of diffusion-based approaches for robust data-driven PDE learning.

Anthony Zhou, Amir Barati Farimani · cmu

Neural solvers for partial differential equations (PDEs) have great potential to generate fast and accurate physics solutions, yet their practicality is currently limited by their generalizability. PDEs evolve over broad scales and exhibit diverse behaviors; predicting these phenomena will require learning representations across a wide variety of inputs which may encompass different coefficients, boundary conditions, resolutions, or even equations. As a step towards generalizable PDE modeling, we adapt masked pretraining for physics problems. Through self-supervised learning across PDEs, masked autoencoders can consolidate heterogeneous physics to learn rich latent representations. We show that learned representations can generalize to a limited set of unseen equations or parameters and are meaningful enough to regress PDE coefficients or the classify PDE features. Furthermore, conditioning neural solvers on learned latent representations can improve time-stepping and super-resolution performance across a variety of coefficients, discretizations, or boundary conditions, as well as on certain unseen PDEs. We hope that masked pretraining can emerge as a unifying method across large, unlabeled, and heterogeneous datasets to learn latent physics at scale.

Zijie Li, Anthony Zhou, Saurabh Patil et al. · cmu

Accurate weather forecasting is crucial in various sectors, impacting decision-making processes and societal events. Data-driven approaches based on machine learning models have recently emerged as a promising alternative to numerical weather prediction models given their potential to capture physics of different scales from historical data and the significantly lower computational cost during the prediction stage. Renowned for its state-of-the-art performance across diverse domains, the Transformer model has also gained popularity in machine learning weather prediction. Yet applying Transformer architectures to weather forecasting, particularly on a global scale is computationally challenging due to the quadratic complexity of attention and the quadratic increase in spatial points as resolution increases. In this work, we propose a factorized-attention-based model tailored for spherical geometries to mitigate this issue. More specifically, it utilizes multi-dimensional factorized kernels that convolve over different axes where the computational complexity of the kernel is only quadratic to the axial resolution instead of overall resolution. The deterministic forecasting accuracy of the proposed model on $1.5^\circ$ and 0-7 days' lead time is on par with state-of-the-art purely data-driven machine learning weather prediction models. We also showcase the proposed model holds great potential to push forward the Pareto front of accuracy-efficiency for Transformer weather models, where it can achieve better accuracy with less computational cost compared to Transformer based models with standard attention.

Anthony Zhou, Amir Barati Farimani · cmu

Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a black box by directly predicting the next state. While this is a natural and easy framework for applying neural surrogates, it can be an over-simplified and rigid framework for predicting physics. In this work, we evaluate an alternate framework in which neural solvers predict the temporal derivative and an ODE integrator forwards the solution in time, which has little overhead and is broadly applicable across model architectures and PDEs. We find that by simply changing the training target and introducing numerical integration during inference, neural surrogates can gain accuracy and stability in finely-discretized regimes. Predicting temporal derivatives also allows models to not be constrained to a specific temporal discretization, allowing for flexible time-stepping during inference or training on higher-resolution PDE data. Lastly, we investigate why this framework can be beneficial and in what situations does it work well.

Anthony Zhou, Alexander Wikner, Amaury Lancelin et al. · cmu

Generative models have recently emerged as powerful surrogates for physical systems, demonstrating increased accuracy, stability, and/or statistical fidelity. Most approaches rely on iteratively denoising a Gaussian, a choice that may not be the most effective for autoregressive prediction tasks in PDEs and dynamical systems such as climate. In this work, we benchmark generative models across diverse physical domains and tasks, and highlight the role of stochastic interpolants. By directly learning a stochastic process between current and future states, stochastic interpolants can leverage the proximity of successive physical distributions. This allows for generative models that can use fewer sampling steps and produce more accurate predictions than models relying on transporting Gaussian noise. Our experiments suggest that generative models need to balance deterministic accuracy, spectral consistency, and probabilistic calibration, and that stochastic interpolants can potentially fulfill these requirements by adjusting their sampling. This study establishes stochastic interpolants as a competitive baseline for physical emulation and gives insight into the abilities of different generative modeling frameworks.

Anthony Zhou, Amir Barati Farimani · cmu

Designing neural networks within a Hamiltonian framework offers a principled way to ensure that conservation laws are respected in physical systems. While promising, these capabilities have been largely limited to discrete, analytically solvable systems. In contrast, many physical phenomena are governed by PDEs, which govern infinite-dimensional fields through Hamiltonian functionals and their functional derivatives. Building on prior work, we represent the Hamiltonian functional as a kernel integral parameterized by a neural field, enabling learnable function-to-scalar mappings and the use of automatic differentiation to calculate functional derivatives. This allows for an extension of Hamiltonian mechanics to neural PDE solvers by predicting a functional and learning in the gradient domain. We show that the resulting Hamiltonian Neural Solver (HNS) can be an effective surrogate model through improved stability and conserving energy-like quantities across 1D and 2D PDEs. This ability to respect conservation laws also allows HNS models to better generalize to longer time horizons or unseen initial conditions.

Anthony Zhou, Cooper Lorsung, AmirPouya Hemmasian et al.

Pretraining for partial differential equation (PDE) modeling has recently shown promise in scaling neural operators across datasets to improve generalizability and performance. Despite these advances, our understanding of how pretraining affects neural operators is still limited; studies generally propose tailored architectures and datasets that make it challenging to compare or examine different pretraining frameworks. To address this, we compare various pretraining methods without optimizing architecture choices to characterize pretraining dynamics on different models and datasets as well as to understand its scaling and generalization behavior. We find that pretraining is highly dependent on model and dataset choices, but in general transfer learning or physics-based pretraining strategies work best. In addition, pretraining performance can be further improved by using data augmentations. Lastly, pretraining can be additionally beneficial when fine-tuning in scarce data regimes or when generalizing to downstream data similar to the pretraining distribution. Through providing insights into pretraining neural operators for physics prediction, we hope to motivate future work in developing and evaluating pretraining methods for PDEs.