TL;DR
This paper introduces SINO, a spectral-inspired neural operator capable of learning complex PDE dynamics from limited data without explicit physics, achieving state-of-the-art accuracy and out-of-distribution generalization.
Contribution
SINO is a novel physics-agnostic neural operator that captures local and global derivatives from spectral features, enabling effective learning from very limited data.
Findings
SINO outperforms existing methods by 1-2 orders of magnitude in accuracy.
SINO requires only 2-5 trajectories for training.
SINO generalizes well to out-of-distribution cases.
Abstract
Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability. To address these challenges, we propose Spectral-Inspired Neural Operator (SINO), which can model complex systems from just 2-5 trajectories, without requiring explicit PDE terms. Specifically, SINO automatically captures both local and global spatial derivatives from frequency indices, enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing. Extensive experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves…
Peer Reviews
Decision·Submitted to ICLR 2026
The main strength of this paper lies in its data efficiency and physics-inspired inductive bias. By operating in the frequency domain and embedding derivative-like interactions through the pi block, the model implicitly encodes structural knowledge of PDE dynamics while remaining fully data-driven. This architectural design allows SINO to achieve remarkable performance using only a handful of training trajectories, outperforming other data-driven solvers that require hundreds or thousands of sam
Despite its strong empirical results, the paper has several limitations that should be addressed. First, the interpretability claims are primarily qualitative: while Figure 5 suggests that the Π-block features align with ground-truth differential operators, the paper provides no quantitative metrics (e.g., correlation or projection scores) to substantiate this claim. This weakens the argument that SINO learns physically meaningful spectral structures. Second, the 2/3 de-aliasing rule is fixed th
- The authors propose an interesting surrogate architecture that directly mimics spectral numerical solvers. The inclusion of classical ingredients such as the 2/3 de-aliasing rule and Runge–Kutta-4 integration demonstrates a clear attempt to build numerical stability into the rollout process, something that many prior neural PDE solvers neglect. - By structuring the model around Fourier transforms and multiplicative nonlinear interactions, SINO incorporates strong physics-motivated priors (e.g
- Clarity and Notation The paper is poorly written and the notation is vague, making it difficult to follow the technical exposition. Section 3.3 (Architecture components), in particular, is imprecise and lacks formal definitions for core modules. - Architectural Novelty Is Overstated The “Spectral Learning Block” is highly similar to the Fourier layer in the Fourier Neural Operator (FNO). Both perform Fourier transforms, apply learnable frequency-domain multipliers, and inverse-transform the
- The OOD generalization and super-resolution experiments demonstrate that SINO is able to approximate the underlying operator using only a very limited number of trajectories. - The architecture is motivated by first principles, resulting in a learnable (physics-agnostic) version of classical spectral solvers and providing a good inductive bias for low-data regimes. - The paper provides ablation studies confirming the necessity of each key component.
There are two main concerns: 1. In almost all experiments, the parameter count of the baselines is significantly larger than the one of SINO. Together with the relatively small grid of hyperparameters used for each method, it seems that all these methods are overfitting to the limited amount of training data. For a fair comparison, each baseline should be evaluated with several configurations that lead to a comparable parameter count. Moreover, it is unclear how much samples from each trajector
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