Fourier-MIONet: Neural Operator for GCS

• The paper introduces Fourier-MIONet, a neural surrogate model that embeds Fourier Neural Operator layers within a multi-input framework to map complex inputs to nonlinear PDE solutions.
• It employs dedicated branch and trunk networks with temporal inputs and U-FNO decoding to efficiently capture high-dimensional parameter-to-solution mappings for CO2 and brine flow.
• The study shows significant improvements with about 3.7M parameters, 3.5× faster training, and enhanced physical consistency over 30-year simulations compared to traditional models.

Fourier-MIONet is a Fourier-enhanced multiple-input neural operator (MIONet) designed as a neural surrogate model for multiphase flow in porous media, particularly for simulating the evolution of CO$_2$ and brine in geological carbon sequestration (GCS). It achieves significant computational efficiency and generalization advantages by embedding the Fourier Neural Operator (FNO) architecture within a multiple-input operator learning framework, enabling rapid, data-efficient, and physically consistent predictions of high-dimensional parameter-to-solution maps governed by nonlinear partial differential equations (PDEs) (Jiang et al., 2023).

1. Mathematical Formulation and Operator Learning Framework

Fourier-MIONet is constructed as a parametric operator learning architecture aimed at mapping multiple spatial and scalar input fields to solution fields of interest, such as CO$_2$ saturation and pressure. Given $n$ input fields $a_1, a_2, \dots, a_n$ over corresponding domains $D_i$, the goal is to approximate the solution operator $\mathcal{G}$ such that

$\mathcal{G}:(a_1, a_2, ..., a_n) \mapsto u,$

where $u$ is a field over the target domain $D'$. This is achieved through the MIONet structure, where $n$ branch networks $b^i_j(a_i)$ encode each input, and a trunk network $t_j(\xi)$ encodes target coordinates $\xi$: $$\mathcal{G}(a_1,\dots,a_n)(\xi) = \sum_{j=1}^p\left(\prod_{i=1}^n b^i_j(a_i)\right) t_j(\xi) + b_0.$$ Temporal continuity is enforced by treating time $t$ as an input to a trunk net, promoting physically consistent evolution across $t$ for $u(x, y, t)$.

2. Network Architecture and Layer Design

Fourier-MIONet integrates FNO layers for learning global, mesh-free operator representations and U-FNO decoding to enhance expressive capacity:

• Branch Networks:
  • Field branch: 2D convolutional encoder with three $3\times3$ convolution layers (20 channels each, ReLU activations, paddings), producing latent tensors of shape (batch, $M_m$, $M_m$) with $M_m=12$ retained Fourier modes.
  • Scalar branch: Four-layer multilayer perceptron (MLP) with widths [10, 20, 20, 20], mapping scalars to (batch, $12 \times 12$).
• Trunk Net: Three-layer MLP (widths 1→20→20→20) taking time $t$ and returning (batch, $12 \times 12$).
• Fusion Operations:
  • Branch merge: elementwise sum $\mathbf{b} = \mathbf{b}_1+\mathbf{b}_2$
  • Branch–trunk merge: elementwise product $z = \mathbf{b} \odot \mathbf{c}$, yielding $z(x, y)$ for each time step
• Decoder (U-FNO): Four alternating FNO and U-Fourier layers in 2D, with each U-Fourier layer augmented by a small 2D U-Net. The decoder concludes with a $1 \times 1$ convolution to map to the output channel.
• Parameter Count: Approximately $3.7 \times 10^6$ trainable parameters—one order-of-magnitude fewer than U-FNO's $33 \times 10^6$.

FNO layers exploit the efficiency of the discrete Fourier transform: for a latent tensor $v(x)$, modes $\hat v(k) = \mathcal{F}[v](k)$ are processed with learned multipliers $\mathcal{R}(k)$ (acting on low-frequency modes), with the spatial representation recovered by inverse transform and pointwise linear-activation additions: $$z_{j+1}(x) = \sigma\left(\mathcal{F}^{-1}[\mathcal{R}_j\,\mathcal{F}(z_j)](x) + W_j z_j(x) + b_j \right).$$

3. Governing Physical Model and Problem Setting

Fourier-MIONet addresses the nonlinear coupled PDE system describing two-phase (CO$_2$/water) flow in porous geologic formations:

• Mass conservation: For phase $i\in\{\mathrm{w}, \mathrm{n}\}$,

$$\phi(x) \frac{\partial S_i}{\partial t} + \nabla \cdot \mathbf{q}_i = 0,$$

where $\phi$ is porosity, $S_i$ is phase saturation.

• Phase flux (Darcy’s Law):

$$\mathbf{q}_i = -\frac{k(x)k_{r,i}(S)}{\mu_i}(\nabla p_i - \rho_i g \nabla z),$$

with absolute permeability $k(x)$, relative permeability $k_{r,i}(S)$, viscosity $\mu_i$, and other standard parameters.

• Capillary pressure coupling: $p_\mathrm{n} = p_\mathrm{w} + p_c(S)$.
• Constitutive relations: $k_{r,i}(S)$ and $p_c(S)$ follow van Genuchten or Brooks–Corey models.

This formulation supports a wide range of input heterogeneity—including anisotropic and highly variable permeability fields, variable injection configurations, and process parameters—requiring an operator learning approach with strong generalization characteristics.

4. Training Regime, Resource Efficiency, and Performance

The network is trained on simulated datasets comprising 4,500 training instances and 500 test cases, with inputs discretized on a $96 \times 200$ grid and 24 logarithmically spaced time snapshots ($1$ day to $30$ years):

• Inputs: 2D spatial fields ($k_x, k_y, \phi$), scalar parameters (initial pressure $P_{\rm init}$, temperature $T$, injection rate $Q$, viscosity ratio $\lambda$, irreducible water saturation $S_{wi}$, perforation depths).
• Loss Function: $L_p$ loss as in U-FNO,

$$L(y, \hat y) = \frac{\|y - \hat y\|_2}{\|y\|_2} + 0.5 \frac{\|\partial_r y - \partial_r \hat y\|_2}{\|\partial_r y\|_2},$$

balancing fieldwise error and spatial gradient error.

• Optimization: Adam optimizer (initial learning rate $10^{-3}$ with decay). Batch size: $4 \times 8$ (cases $\times$ time snapshots) per update.
• Resource Usage Comparison:

Model	Params	CPU Mem (GiB)	GPU Mem (GiB)	Training Time	Inference (s/case)	Speedup
U-FNO	$33$M	$103$	$15.9$	$48$ h	$0.075$	1$\times$
Fourier-MIONet	$3.7$M	$15$	$12.8$	$14$ h	$0.041$	$3.5\times$ (train), $1.8\times$ (infer)

• Test Accuracy: $R^2 \approx 0.984$, MAE $\approx 4.6\times10^{-3}$, matching U-FNO.

5. Generalization, Data Efficiency, and Physical Consistency

Fourier-MIONet explicitly encodes temporal continuity in the operator by treating time $t$ as an input coordinate to the trunk network rather than as a discrete channel. This enables the model to interpolate and extrapolate $u(x, t)$ smoothly over time, allowing accurate forecasting of the full 30-year CO$_2$ evolution from as few as 6 (nonuniformly chosen) training snapshots (test $R^2 \sim 0.98$, MAE $\sim 4 \times 10^{-3}$). This is in contrast to architectures relying on fixed time-channel discretization (e.g., U-FNO), which are unable to generalize to intermediate or unseen time points unless many snapshots are observed in training.

The approach has been validated under demanding scenarios, including permeability heterogeneity spanning four orders of magnitude, anisotropy ratios up to 10, and wide ranges of injection rates and capillary scales, consistently maintaining robust performance (Jiang et al., 2023).

6. Advantages, Limitations, and Practical Considerations

Advantages:

• One order-of-magnitude fewer parameters than U-FNO while maintaining accuracy.
• $\sim3.5\times$ reduction in training time, $\sim1.8\times$ faster inference.
• Substantially reduced memory footprint: 85% less CPU memory, 64% less GPU memory.
• Extreme data efficiency, supporting accurate 30-year predictions from only six data snapshots.
• Intrinsically enforces temporal continuity and physical consistency.

Limitations:

• The architecture relies on regular spatial grids to facilitate the FFT; extension to unstructured meshes is nontrivial.
• Fourier convolution’s global nature may impact scalability in extremely large or high-dimensional spatial domains.
• Hyperparameter selection (e.g., number of Fourier modes, network widths) remains application-specific.

A plausible implication is that, as with FNO-based models generally, pushing Fourier-MIONet toward very large or irregular domains may necessitate new domain decomposition or hybrid mesh strategies.

7. Broader Context and Comparison with Related Architectures

Fourier-MIONet exemplifies the integration of operator learning (MIONet) and Fourier-domain modeling (FNO), aligning with recent advances in mesh-free, global-receptive-field deep architectures for scientific machine learning. In operator learning, FNO layers enable efficient learning of parametric solution maps for PDE-governed systems, and the multi-input/trunk design allows explicit inclusion of both field and scalar process variables.

The architectural principles of Fourier-MIONet are applicable to a broad class of problems requiring solution of high-dimensional, parameterized PDEs—extending beyond geological carbon sequestration to possible applications in subsurface hydrology, reactive transport, and time-continuous simulation surrogates (Jiang et al., 2023).

Fourier neural operators have also begun to demonstrate potential in physical-layer modeling for complex electromagnetic and communications systems, highlighting the generality of the core convolutional operator-learning paradigm (Xiao et al., 6 Oct 2025). However, Fourier-MIONet's contributions are characterized by its tailored multi-input framework, physical consistency over time, and demonstrated empirical gains in data efficiency and computational resource usage for carbon storage simulation.