CoNO: Complex Neural Operator for Continous Dynamical Physical Systems

TL;DR

CoNO tackles the challenge of learning solution operators for non-stationary PDEs by merging complex-valued neural networks with a Fractional Fourier Transform–based kernel. The approach provides a universal approximation framework and demonstrates state-of-the-art performance across seven PDE benchmarks, including robustness to noise, zero-shot super-resolution, and data-efficient training. Empirically, CoNO yields significant gains on time-dependent problems and maintains stability across resolutions, suggesting strong potential for real-time PDE inference in scientific computing. The combination of FrFT-based kernels and CVNNs offers a principled path to model non-stationary dynamics in a continuous-to-discrete operator learning setting.

Abstract

Neural operators extend data-driven models to map between infinite-dimensional functional spaces. While these operators perform effectively in either the time or frequency domain, their performance may be limited when applied to non-stationary spatial or temporal signals whose frequency characteristics change with time. Here, we introduce Complex Neural Operator (CoNO) that parameterizes the integral kernel using Fractional Fourier Transform (FrFT), better representing non-stationary signals in a complex-valued domain. Theoretically, we prove the universal approximation capability of CoNO. We perform an extensive empirical evaluation of CoNO on seven challenging partial differential equations (PDEs), including regular grids, structured meshes, and point clouds. Empirically, CoNO consistently attains state-of-the-art performance, showcasing an average relative gain of 10.9%. Further, CoNO exhibits superior performance, outperforming all other models in additional tasks such as zero-shot super-resolution and robustness to noise. CoNO also exhibits the ability to learn from small amounts of data -- giving the same performance as the next best model with just 60% of the training data. Altogether, CoNO presents a robust and superior model for modeling continuous dynamical systems, providing a fillip to scientific machine learning.

Figures (10)

• Figure 1: FrFT heatmaps illustrating the temporal-frequency characteristics of the 2D Navier-Stokes equation for varying values of $\alpha$. Each subplot represents the magnitude of the transformed frequency content over time, obtained by applying the FrFT and then flattened 2D frequency map of the Navier-Stokes equation. Different subplots correspond to fractional orders $\alpha$, highlighting the diverse spectral behaviors captured by the FrFT across both the temporal and frequency domains. Note that $\alpha = 0$ represents the time domain while $\alpha = 1$ represents the frequency domain.
• Figure 2: CoNO Architecture Overview.(Top) (1) The input function $a(x)$ undergoes a deformation $\phi^{-1}$ to convert an irregular mesh into a uniform mesh. (2) The deformed input is then lifted to a higher dimension in the channel space using a neural network. (3) Apply iterative CoNO layer in complex domain consisting of fractional integral kernel. (4) Then, the output is projected back on the lower dimension in channel space using neural network. (5) Solution $u(x)$ is obtained by passing through the deformation $\phi$. (Bottom) Zoomed version of FrFT integral kernel defined in Eq. \ref{['equ: non linear operator']} with learnable parameters $R^{\alpha}$, $R^{\alpha'}$, $W$ and fractional order $\alpha$ and $\alpha'$.
• Figure 3: Depiction of results for different methods on fluid datasets, Darcy Flow (Left) and Navier Stokes (Right). We plotted the heatmap of the absolute difference value between ground truth and prediction to compare the predicted output. See the App. Fig. \ref{['fig:figure10']} for more solid physics and fluid physics benchmarks showcases.
• Figure 4: (Left) Models performance under different resolutions on Darcy. (Middle) Models performance under different training datasets the ratio on Darcy. (Right) Models performance under the presence of noise on Darcy. The lower $l2$ loss indicates better performance.
• Figure 5: Learning Curve for Darcy flow (Left) and Navier Stokes (Right) where the x-axis denotes epochs and the y-axis $l2$ error.

Theorems & Definitions (21)

• Definition 3.1: FrFT
• Remark 3.2
• Definition 3.3: Complex Valued Activation
• Theorem 4.1: Product Rule for FrFT
• Theorem 4.2: Convolution Theorem for FrFT
• Theorem 4.3: Universal Approximation
• Definition B.1: Fractional Torus
• Definition B.2
• Definition B.3
• Definition B.4: Fractional Fourier Transform on Fractional Torus fu2022convergence