GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning

Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle to capture this variability. To address this, data-driven approaches learn parametric PDEs by sampling a very large variety of trajectories with varying PDE parameters. We first show that incorporating conditioning mechanisms for learning parametric PDEs is essential and that among them, $\textit{adaptive conditioning}$, allows stronger generalization. As existing adaptive conditioning methods do not scale well with respect to the number of parameters to adapt in the neural solver, we propose GEPS, a simple adaptation mechanism to boost GEneralization in Pde Solvers via a first-order optimization and low-rank rapid adaptation of a small set of context parameters. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers. Validation performed on a whole range of spatio-temporal forecasting problems demonstrates excellent performance for generalizing to unseen conditions including initial conditions, PDE coefficients, forcing terms and solution domain. $\textit{Project page}$: https://geps-project.github.io

Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem, caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach addresses parametric PDEs. Specifically, our method integrates the physical loss gradient with the PDE parameters to solve over a distribution of PDE parameters, including coefficients, initial conditions, or boundary conditions. We demonstrate the effectiveness of our method through empirical experiments on multiple datasets, comparing training and test-time optimization performance.

Probing Language Models on Their Knowledge Source

Large Language Models (LLMs) often encounter conflicts between their learned, internal (parametric knowledge, PK) and external knowledge provided during inference (contextual knowledge, CK). Understanding how LLMs models prioritize one knowledge source over the other remains a challenge. In this paper, we propose a novel probing framework to explore the mechanisms governing the selection between PK and CK in LLMs. Using controlled prompts designed to contradict the model's PK, we demonstrate that specific model activations are indicative of the knowledge source employed. We evaluate this framework on various LLMs of different sizes and demonstrate that mid-layer activations, particularly those related to relations in the input, are crucial in predicting knowledge source selection, paving the way for more reliable models capable of handling knowledge conflicts effectively.

Zebra: In-Context and Generative Pretraining for Solving Parametric PDEs

Solving time-dependent parametric partial differential equations (PDEs) is challenging, as models must adapt to variations in parameters such as coefficients, forcing terms, and boundary conditions. Data-driven neural solvers either train on data sampled from the PDE parameters distribution in the hope that the model generalizes to new instances or rely on gradient-based adaptation and meta-learning to implicitly encode the dynamics from observations. This often comes with increased inference complexity. Inspired by the in-context learning capabilities of large language models (LLMs), we introduce Zebra, a novel generative auto-regressive transformer designed to solve parametric PDEs without requiring gradient adaptation at inference. By leveraging in-context information during both pre-training and inference, Zebra dynamically adapts to new tasks by conditioning on input sequences that incorporate context trajectories or preceding states. This approach enables Zebra to flexibly handle arbitrarily sized context inputs and supports uncertainty quantification through the sampling of multiple solution trajectories. We evaluate Zebra across a variety of challenging PDE scenarios, demonstrating its adaptability, robustness, and superior performance compared to existing approaches.

NeurIPS 2024 ML4CFD Competition: Harnessing Machine Learning for Computational Fluid Dynamics in Airfoil Design

The integration of machine learning (ML) techniques for addressing intricate physics problems is increasingly recognized as a promising avenue for expediting simulations. However, assessing ML-derived physical models poses a significant challenge for their adoption within industrial contexts. This competition is designed to promote the development of innovative ML approaches for tackling physical challenges, leveraging our recently introduced unified evaluation framework known as Learning Industrial Physical Simulations (LIPS). Building upon the preliminary edition held from November 2023 to March 2024, this iteration centers on a task fundamental to a well-established physical application: airfoil design simulation, utilizing our proposed AirfRANS dataset. The competition evaluates solutions based on various criteria encompassing ML accuracy, computational efficiency, Out-Of-Distribution performance, and adherence to physical principles. Notably, this competition represents a pioneering effort in exploring ML-driven surrogate methods aimed at optimizing the trade-off between computational efficiency and accuracy in physical simulations. Hosted on the Codabench platform, the competition offers online training and evaluation for all participating solutions.

AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields

We present AROMA (Attentive Reduced Order Model with Attention), a framework designed to enhance the modeling of partial differential equations (PDEs) using local neural fields. Our flexible encoder-decoder architecture can obtain smooth latent representations of spatial physical fields from a variety of data types, including irregular-grid inputs and point clouds. This versatility eliminates the need for patching and allows efficient processing of diverse geometries. The sequential nature of our latent representation can be interpreted spatially and permits the use of a conditional transformer for modeling the temporal dynamics of PDEs. By employing a diffusion-based formulation, we achieve greater stability and enable longer rollouts compared to conventional MSE training. AROMA's superior performance in simulating 1D and 2D equations underscores the efficacy of our approach in capturing complex dynamical behaviors.

ML4PhySim : Machine Learning for Physical Simulations Challenge (The airfoil design)

The use of machine learning (ML) techniques to solve complex physical problems has been considered recently as a promising approach. However, the evaluation of such learned physical models remains an important issue for industrial use. The aim of this competition is to encourage the development of new ML techniques to solve physical problems using a unified evaluation framework proposed recently, called Learning Industrial Physical Simulations (LIPS). We propose learning a task representing a well-known physical use case: the airfoil design simulation, using a dataset called AirfRANS. The global score calculated for each submitted solution is based on three main categories of criteria covering different aspects, namely: ML-related, Out-Of-Distribution, and physical compliance criteria. To the best of our knowledge, this is the first competition addressing the use of ML-based surrogate approaches to improve the trade-off computational cost/accuracy of physical simulation.The competition is hosted by the Codabench platform with online training and evaluation of all submitted solutions.

LOCOST: State-Space Models for Long Document Abstractive Summarization

State-space models are a low-complexity alternative to transformers for encoding long sequences and capturing long-term dependencies. We propose LOCOST: an encoder-decoder architecture based on state-space models for conditional text generation with long context inputs. With a computational complexity of $O(L \log L)$, this architecture can handle significantly longer sequences than state-of-the-art models that are based on sparse attention patterns. We evaluate our model on a series of long document abstractive summarization tasks. The model reaches a performance level that is 93-96% comparable to the top-performing sparse transformers of the same size while saving up to 50% memory during training and up to 87% during inference. Additionally, LOCOST effectively handles input texts exceeding 600K tokens at inference time, setting new state-of-the-art results on full-book summarization and opening new perspectives for long input processing.

Module-wise Training of Neural Networks via the Minimizing Movement Scheme

Greedy layer-wise or module-wise training of neural networks is compelling in constrained and on-device settings where memory is limited, as it circumvents a number of problems of end-to-end back-propagation. However, it suffers from a stagnation problem, whereby early layers overfit and deeper layers stop increasing the test accuracy after a certain depth. We propose to solve this issue by introducing a module-wise regularization inspired by the minimizing movement scheme for gradient flows in distribution space. We call the method TRGL for Transport Regularized Greedy Learning and study it theoretically, proving that it leads to greedy modules that are regular and that progressively solve the task. Experimentally, we show improved accuracy of module-wise training of various architectures such as ResNets, Transformers and VGG, when our regularization is added, superior to that of other module-wise training methods and often to end-to-end training, with as much as 60% less memory usage.

INFINITY: Neural Field Modeling for Reynolds-Averaged Navier-Stokes Equations

For numerical design, the development of efficient and accurate surrogate models is paramount. They allow us to approximate complex physical phenomena, thereby reducing the computational burden of direct numerical simulations. We propose INFINITY, a deep learning model that utilizes implicit neural representations (INRs) to address this challenge. Our framework encodes geometric information and physical fields into compact representations and learns a mapping between them to infer the physical fields. We use an airfoil design optimization problem as an example task and we evaluate our approach on the challenging AirfRANS dataset, which closely resembles real-world industrial use-cases. The experimental results demonstrate that our framework achieves state-of-the-art performance by accurately inferring physical fields throughout the volume and surface. Additionally we demonstrate its applicability in contexts such as design exploration and shape optimization: our model can correctly predict drag and lift coefficients while adhering to the equations.