Convolution Operator Network for Forward and Inverse Problems (FI-Conv): Application to Plasma Turbulence Simulations

TL;DR

FI-Conv introduces a Convolutional Operator Network that unifies forward prediction and inverse PDE parameter estimation within a single, autoregressive framework. By embedding initial state, PDE parameters, and evolution time into a U-Net equipped with ConvNeXt V2 blocks, the method achieves accurate short-term forecasts and statistically faithful long-term behavior for turbulent HW plasma dynamics, while enabling gradient-based parameter inference without retraining. The approach is demonstrated on a multi-parameter HW model, showing competitive forward accuracy, preservation of high-frequency structures, and effective inverse estimation for multiple parameters, with k0 and cpb being most readily inferred. The results suggest FI-Conv as a scalable, data-driven surrogate for complex, multiscale PDEs and a practical tool for plasma diagnostics and optimization in tokamak-like systems.

Abstract

We propose the Convolutional Operator Network for Forward and Inverse Problems (FI-Conv), a framework capable of predicting system evolution and estimating parameters in complex spatio-temporal dynamics, such as turbulence. FI-Conv is built on a U-Net architecture, in which most convolutional layers are replaced by ConvNeXt V2 blocks. This design preserves U-Net performance on inputs with high-frequency variations while maintaining low computational complexity. FI-Conv uses an initial state, PDE parameters, and evolution time as input to predict the system future state. As a representative example of a system exhibiting complex dynamics, we evaluate the performance of FI-Conv on the task of predicting turbulent plasma fields governed by the Hasegawa-Wakatani (HW) equations. The HW system models two-dimensional electrostatic drift-wave turbulence and exhibits strongly nonlinear behavior, making accurate approximation and long-term prediction particularly challenging. Using an autoregressive forecasting procedure, FI-Conv achieves accurate forward prediction of the plasma state evolution over short times (t ~ 3) and captures the statistic properties of derived physical quantities of interest over longer times (t ~ 100). Moreover, we develop a gradient-descent-based inverse estimation method that accurately infers PDE parameters from plasma state evolution data, without modifying the trained model weights. Collectively, our results demonstrate that FI-Conv can be an effective alternative to existing physics-informed machine learning methods for systems with complex spatio-temporal dynamics.

Figures (12)

• Figure 1: Schematic of the proposed FI-Conv architecture. ConvNeXt V2 blocks are shown in green (left). The input field and embedded parameters, including the evolution time and PDE parameters, are shown in blue. The first, second, third, and fourth depth levels of the U-Net encoder–decoder hierarchy are indicated in red, brown, yellow, and gray, respectively. The enforcement of hard initial-condition constraints is highlighted in purple.
• Figure 2: Overview of the FI-Conv workflow. Blue arrows denote experimental or numerical simulation data, red arrows indicate FI-Conv forward prediction, yellow arrows represent the training process, and purple arrows correspond to the inverse parameter estimation problem.
• Figure 3: Effect of parameter variations on plasma dynamics on the vorticity $\Omega$. From left to right, the varied parameters are $c_1$, $k_0$, $\kappa$, and $c_{pb}$. Variations in different parameters lead to qualitatively distinct dynamical behaviors. This motivates the inclusion of all parameters in the FI-Conv experiments.
• Figure 4: One-step prediction of the plasma state at 0.8 time units after an input initial state in the testing data. Left column: reference HW2D simulation code results. Middle column: FI-Conv prediction. Right column: difference between the HW2D results and the FI-Conv prediction. The time after the FI-Conv input state is $\Delta t_{\mathrm{i}}=0.8$.
• Figure 5: Autoregressive prediction of plasma state with time step $t_{\mathrm{a}} = 0.75$ from an input initial state in the testing data. Left column: reference $\Omega$ snapshots computed with the HW2D simulation code. Middle column: $\Omega$ snapshots predicted by FI-Conv after 4, 8, and 16 autoregressive steps. Right column: difference between the HW2D results and the FI-Conv prediction. From upper to lower, the time after the FI-Conv input state $\Delta t_{\mathrm{i}}$ is 3.0, 6.0, and 12.0, respectively.