Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Limits of Resolution Equivariance in Fourier Neural Operators

Fourier Neural Operators are often assumed to generalize across spatial resolutions, enabling training on a coarse grid and deployment on a finer grid. We test this assumption by contrasting two inference-time choices when moving from training resolution $s$ to test resolution $S>s$: running FNO directly at $S$, or running at $s$ and upsampling the prediction to $S$ via Fourier zero-padding. On Darcy flow, we observe that direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. We further analyze layerwise spectra and find that, under Fourier truncation, intermediate representations increasingly concentrate energy in low frequencies, with high-frequency output produced mainly by late nonlinear/decoder stages. This offers a mechanistic explanation for why FNO can perform well while retaining few modes, yet remain sensitive under resolution shifts. Our findings highlight a simple but strong baseline for cross-resolution evaluation and point to nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.

Polynomial Mixing for Efficient Self-supervised Speech Encoders

State-of-the-art speech-to-text models typically employ Transformer-based encoders that model token dependencies via self-attention mechanisms. However, the quadratic complexity of self-attention in both memory and computation imposes significant constraints on scalability. In this work, we propose a novel token-mixing mechanism, the Polynomial Mixer (PoM), as a drop-in replacement for multi-head self-attention. PoM computes a polynomial representation of the input with linear complexity with respect to the input sequence length. We integrate PoM into a self-supervised speech representation learning framework based on BEST-RQ and evaluate its performance on downstream speech recognition tasks. Experimental results demonstrate that PoM achieves a competitive word error rate compared to full self-attention and other linear-complexity alternatives, offering an improved trade-off between performance and efficiency in time and memory.

Linear Attention with Global Context: A Multipole Attention Mechanism for Vision and Physics

Transformers have become the de facto standard for a wide range of tasks, from image classification to physics simulations. Despite their impressive performance, the quadratic complexity of standard Transformers in both memory and time with respect to the input length makes them impractical for processing high-resolution inputs. Therefore, several variants have been proposed, the most successful relying on patchification, downsampling, or coarsening techniques, often at the cost of losing the finest-scale details. In this work, we take a different approach. Inspired by state-of-the-art techniques in $n$-body numerical simulations, we cast attention as an interaction problem between grid points. We introduce the Multipole Attention Neural Operator (MANO), which computes attention in a distance-based multiscale fashion. MANO maintains, in each attention head, a global receptive field and achieves linear time and memory complexity with respect to the number of grid points. Empirical results on image classification and Darcy flows demonstrate that MANO rivals state-of-the-art models such as ViT and Swin Transformer, while reducing runtime and peak memory usage by orders of magnitude. We open source our code for reproducibility at https://github.com/AlexColagrande/MANO.

Recommendation of data-free class-incremental learning algorithms by simulating future data

Class-incremental learning deals with sequential data streams composed of batches of classes. Various algorithms have been proposed to address the challenging case where samples from past classes cannot be stored. However, selecting an appropriate algorithm for a user-defined setting is an open problem, as the relative performance of these algorithms depends on the incremental settings. To solve this problem, we introduce an algorithm recommendation method that simulates the future data stream. Given an initial set of classes, it leverages generative models to simulate future classes from the same visual domain. We evaluate recent algorithms on the simulated stream and recommend the one which performs best in the user-defined incremental setting. We illustrate the effectiveness of our method on three large datasets using six algorithms and six incremental settings. Our method outperforms competitive baselines, and performance is close to that of an oracle choosing the best algorithm in each setting. This work contributes to facilitate the practical deployment of incremental learning.

An Analysis of Initial Training Strategies for Exemplar-Free Class-Incremental Learning

Class-Incremental Learning (CIL) aims to build classification models from data streams. At each step of the CIL process, new classes must be integrated into the model. Due to catastrophic forgetting, CIL is particularly challenging when examples from past classes cannot be stored, the case on which we focus here. To date, most approaches are based exclusively on the target dataset of the CIL process. However, the use of models pre-trained in a self-supervised way on large amounts of data has recently gained momentum. The initial model of the CIL process may only use the first batch of the target dataset, or also use pre-trained weights obtained on an auxiliary dataset. The choice between these two initial learning strategies can significantly influence the performance of the incremental learning model, but has not yet been studied in depth. Performance is also influenced by the choice of the CIL algorithm, the neural architecture, the nature of the target task, the distribution of classes in the stream and the number of examples available for learning. We conduct a comprehensive experimental study to assess the roles of these factors. We present a statistical analysis framework that quantifies the relative contribution of each factor to incremental performance. Our main finding is that the initial training strategy is the dominant factor influencing the average incremental accuracy, but that the choice of CIL algorithm is more important in preventing forgetting. Based on this analysis, we propose practical recommendations for choosing the right initial training strategy for a given incremental learning use case. These recommendations are intended to facilitate the practical deployment of incremental learning.