Physics-Informed Dual Neural Operator (PIANO)

• Physics-Informed Dual Neural Operator (PIANO) is a framework that integrates neural operator design with physics-based loss terms to ensure both accurate predictions and adherence to physical laws.
• It employs a dual structure with a time-stepping operator and a velocity-extraction operator to enforce PDE constraints in applications like precipitation nowcasting and solar magnetic field extrapolation.
• PIANO demonstrates improved forecasting skill by fusing data-driven learning with explicit physical modeling, ensuring robust, interpretable results in high-dimensional scenarios.

Physics-Informed Dual Neural Operator (PIANO) refers to advanced neural operator models that encode explicit physical constraints within their architecture to learn mappings between complex spatiotemporal fields. Recent formulations of PIANO have been developed for heterogeneous scientific problems, including precipitation nowcasting and solar magnetic field extrapolation. By jointly leveraging physics-based loss terms and neural operator design, PIANO models achieve both accuracy and physical consistency on high-dimensional prediction tasks using sparse, incomplete, or indirect observations.

1. Rationale for Physics-Informed Dual Neural Operators

Classical scientific forecasting—exemplified by numerical weather prediction (NWP)—relies on discretizing governing PDEs and solving them iteratively on large supercomputers, thus limiting availability in resource-constrained or data-poor regions. Purely data-driven deep learning methods can fill coverage gaps, particularly with remote sensing data (e.g., satellite imagery), but they often lack physical plausibility, exhibit poor generalization, and may produce over-smoothed, unphysical predictions.

PIANO frameworks address these limitations by combining neural operator expressiveness with physics-informed regularization. In precipitation nowcasting, for example, PIANO learns to step satellite imagery forward in time through a dual operator structure: one operator learns general spatiotemporal evolution, while the other extracts velocity fields that enforce a discrete advection–diffusion law during training. This explicit physical constraint mitigates pathologies of unconstrained models, improving both accuracy and interpretability (Chin et al., 30 Nov 2025).

Similarly, in solar magnetic field extrapolation, PIANO architectures project 2D surface magnetograms to 3D non-linear force-free field (NLFFF) volumes while explicitly enforcing divergence-free and force-free conditions via dedicated physics loss terms (Cao et al., 6 Oct 2025).

2. Neural Operator Design and Mathematical Formulations

2.1 Precipitation Nowcasting PIANO

For satellite-based precipitation nowcasting, the latent field $u(x,y,t)$ (e.g., infrared brightness temperature) is assumed to satisfy an advection–diffusion PDE: $$\frac{\partial u}{\partial t} = \nabla \cdot (D\nabla u - \mathbf{v}u) + R$$ where $\mathbf{v}$ is a 2D velocity field, $D$ is a diffusion coefficient, and $R$ is a source/sink term.

The neural operator architecture consists of two cascaded components:

• Time-Stepping Neural Operator (T-NO): Predicts future satellite images from recent observations and static context (e.g., digital elevation map). Trained via mean squared error (MSE) loss (data-only).
• Velocity-Extraction Neural Operator (V-NO): Recovers velocity, diffusion, and source fields from the predicted sequence, minimizing the discrete PDE residual via a physics-informed loss.

The training objective is: $$\mathcal{L}_{\rm total} = \mathcal{L}_{\rm data} + \alpha\,\mathcal{L}_{\rm PDE}$$ where $\mathcal{L}_{\rm PDE}$ quantifies violation of the discrete advection–diffusion update, and $\alpha$ controls regularization strength (Chin et al., 30 Nov 2025).

2.2 Solar Magnetic Field Extrapolation PIANO

This variant employs a Fourier Neural Operator backbone with Efficient Channel Attention and Dilated Convolution blocks to learn Fourier-space convolutions efficiently. Inputs include a vector magnetogram and auxiliary physical scalars (domain lengths). Projection and lifting MLP layers convert between input/output space and feature space. The combined prediction loss is: $$\mathcal{L} = \mathcal{L}_{\rm data} + \lambda_{\rm div}\,\mathcal{L}_{\rm div} + \lambda_{\rm ff}\,\mathcal{L}_{\rm ff}$$ where $\mathcal{L}_{\rm div}$ and $\mathcal{L}_{\rm ff}$ penalize divergence and force-free errors on the predicted magnetic field (Cao et al., 6 Oct 2025).

3. Detailed Architecture and Data Flow

Precipitation Nowcasting

• T-NO: Ingests historical satellite frames and static features, outputs predicted frames ($u_{t+1:t+s}$).
• V-NO: Takes predicted frames and static features, inverts the physics to produce velocity ($\mathbf{v}$), diffusion ($D$), and source ($R$) fields, enforcing consistency with observed satellite changes.
• Joint Training: The dual cascade (T-NO→V-NO) enables expressive time prediction unconstrained by physics, with subsequent enforcement of global physical laws.

Solar Field Extrapolation

• Lifting: MLPs lift surface and scalar inputs to a high-dimensional feature space.
• Attention-Enhanced Fourier Layers: Stack of Fourier spectral convolutional layers with pointwise and global attention to model nonlocal spatial relations.
• Projection: Output feature map is projected to the 3D volume, reconstructing $\mathbf{B}_{\mathrm{pred}}$.
• Physics Losses: Enforced during both phases of training to stabilize solution and encourage physical consistency in reconstructed fields.

4. Training Procedures, Datasets, and Losses

Precipitation Nowcasting:

• Dataset: Sat2RDR over South Korea (2020–2024), with IR10.5 µm, WV6.3 µm, WV7.3 µm bands and digital elevation.
• Pre-training: T-NO (data loss), V-NO (PDE loss) on 8h rolling windows.
• Fine-tuning: Cascaded T-NO→V-NO, optimizing $\mathcal{L}_{\rm total}$ on full sequences using Adam (RTX 5090, batch size 3).
• Physics weight ($\alpha$): Ablation revealed optimal $\alpha = 1.0$.

Solar Field Extrapolation:

• Dataset: ISEE NLFFF (2010–2016, 143/27 regions train/test).
• Phased Training: Phase 1 uses only 2D boundary data; Phase 2 fine-tunes using its own 3D predictions as additional input.
• Optimizer: Adam; batch size 1.
• Physics Weights: $\lambda_{\rm div} = 1.0$, $\lambda_{\rm ff} = 0.1$.

5. Evaluation and Benchmarking

Precipitation Nowcasting

Forecast skill is quantified by Critical Success Index (CSI) at thresholds (4 mm/h, 8 mm/h) across 1–8 h horizons. Comparative CSI metrics are summarized below:

Model	CSI 4 mm (1h)	CSI 4 mm (8h)	CSI 8 mm (1h)	CSI 8 mm (8h)
NPM+GAN	0.750	0.755	0.599	0.599
PhyDNet+GAN	0.737	0.763	0.601	0.654
PIANO+GAN	0.757	0.763	0.611	0.614

PIANO outperforms baselines for moderate precipitation at all lead times, shows robust seasonal performance ($\Delta$CSI 4 mm $<$ 0.01 vs. $\sim$0.1 for NPM), and maintains competitive heavy-rain skill out to 8 h (Chin et al., 30 Nov 2025).

Solar Field Extrapolation

Performance on ISEE NLFFF is measured via R², Relative Error (RE), MSE, MAE, PSNR, and SSIM per $\mathbf{B}$ component. For the most challenging B_y component:

Model	R²	RE	MSE	MAE	PSNR	SSIM
PIANO	0.9315	0.2909	0.0685	0.1717	44.86	0.9403
UFNO	0.9310	0.3051	0.0690	0.1799	44.78	0.9347
FNO	0.9178	0.3229	0.0822	0.1904	44.04	0.9246

PIANO achieves top accuracy and physical consistency across all components (Cao et al., 6 Oct 2025).

6. Limitations and Generalization

Precipitation Nowcasting:

• Physics constraints have marginal impact on the first-step prediction due to dominance of data loss.
• Heavy precipitation skill remains below state-of-the-art NWP after ∼2–3 h; further physics (e.g., explicit convection) may be required.
• Requires reliable ancillary data (e.g., DEM) and clear-sky calibration; cloud-induced uncertainty remains challenging.

Generalization:

• The dual-operator structure is agnostic to the underlying PDE and can be adapted for other advection-dominated processes (e.g., ocean wave modeling, pollutant transport, wildfire spread).
• Framework supports various input modalities: multi-spectral satellite, radar, or in situ measurements, depending on application domain.

7. Extensions and Future Directions

• Physics Layer Flexibility: Incorporating learnable or spatially varying diffusion/source fields to model more realistic physical mechanisms, including anisotropy and convection.
• Physics–Data Fusion: Integration with NWP or reanalysis products (e.g., ERA5) yields potential for hybrid modeling approaches, further extending skill at long-range prediction.
• Generative Model Replacement: Upgrading from Pix2Pix GAN to video-focused architectures (e.g., NowcastNet-style ST-GANs) for end-to-end precipitative radar sequence generation.
• Global Deployment: PIANO’s lightweight GPU requirements and reliance on satellite inputs facilitate deployment in data-scarce regions and under-served countries where traditional radar- or simulation-based methods are unavailable.
• Broader Physical Domains: PIANO architectures have been empirically validated for both atmospheric (precipitation) and astrophysical (solar magnetism) systems, suggesting broad applicability for PDE-constrained datadriven problems—conditional on suitable adaptation of network architecture and physics-informed loss design.

In summary, the Physics-Informed Dual Neural Operator framework represents a convergence of operator learning and domain-specific physical modeling, demonstrating improved accuracy, interpretability, and generalization on high-dimensional scientific forecasting tasks compared to both classical and conventional neural operator baselines (Chin et al., 30 Nov 2025, Cao et al., 6 Oct 2025).