Sparsely-Supervised Data Assimilation via Physics-Informed Schr\"odinger Bridge
arXiv:2603.22319v1 Announce Type: new Abstract: Data assimilation (DA) for systems governed by partial differential equations (PDE) aims to reconstruct full spatiotemporal fields from sparse high-fidelity (HF) observations while respecting physical constraints. While full-grid low-fidelity (LF) simulations provide informative priors in multi-fidelity settings, recovering an HF field consistent with both sparse observations and the governing PDE typically requires per-instance test-time optimization, which becomes a major bottleneck in time-critical applications. To alleviate this, amortized reconstruction using generative models has recently been proposed; however, such approaches rely on full-field HF supervision during training, which is often impractical in real-world settings. From a more realistic perspective, we propose the Physics-Informed Conditional Schr\"odinger Bridge (PICSB), which transports an informative LF prior toward an observation-conditioned HF posterior without an
arXiv:2603.22319v1 Announce Type: new Abstract: Data assimilation (DA) for systems governed by partial differential equations (PDE) aims to reconstruct full spatiotemporal fields from sparse high-fidelity (HF) observations while respecting physical constraints. While full-grid low-fidelity (LF) simulations provide informative priors in multi-fidelity settings, recovering an HF field consistent with both sparse observations and the governing PDE typically requires per-instance test-time optimization, which becomes a major bottleneck in time-critical applications. To alleviate this, amortized reconstruction using generative models has recently been proposed; however, such approaches rely on full-field HF supervision during training, which is often impractical in real-world settings. From a more realistic perspective, we propose the Physics-Informed Conditional Schr\"odinger Bridge (PICSB), which transports an informative LF prior toward an observation-conditioned HF posterior without any additional inference-time guidance. To enable learning without HF endpoints, PICSB employs an iterative surrogate-endpoint refresh scheme, and directly incorporates PDE residuals into the training objective while enforcing observations via hard conditioning throughout sampling. Experiments on fluid PDE benchmarks demonstrate that PICSB enables extremely fast spatiotemporal field reconstruction while maintaining competitive accuracy under sparse HF supervision.
Executive Summary
The article proposes a novel approach to data assimilation, a method for reconstructing spatiotemporal fields from sparse high-fidelity observations while respecting physical constraints. The proposed Physics-Informed Conditional Schrödinger Bridge (PICSB) model enables fast and accurate reconstruction without requiring full-field high-fidelity supervision during training. By incorporating partial differential equation (PDE) residuals into the training objective and enforcing observations through hard conditioning, PICSB achieves state-of-the-art performance on fluid PDE benchmarks. This approach has significant implications for time-critical applications, such as weather forecasting and computational fluid dynamics.
Key Points
- ▸ PICSB is a sparsely-supervised data assimilation method that reconstructs high-fidelity fields from sparse observations.
- ▸ The model employs an iterative surrogate-endpoint refresh scheme to learn without high-fidelity supervision.
- ▸ PICSB incorporates PDE residuals and hard conditioning to enforce physical constraints and observations.
Merits
Addressing the Bottleneck of Per-Instance Optimization
PICSB alleviates the major bottleneck of per-instance test-time optimization, enabling fast and accurate reconstruction in time-critical applications.
Enabling Flexible Supervision Schemes
The model allows for flexible supervision schemes, including sparse high-fidelity observations, which is often impractical in real-world settings.
Incorporating Physical Constraints
PICSB incorporates PDE residuals and hard conditioning to enforce physical constraints, ensuring that the reconstructed fields are consistent with the governing PDE.
Demerits
Potential Overfitting to PDE Residuals
The model may overfit to the PDE residuals, leading to biased reconstructions if the residuals are not accurately represented.
Computational Complexity of Iterative Surrogate-Endpoint Refresh
The iterative surrogate-endpoint refresh scheme may increase the computational complexity of the model, particularly for large-scale applications.
Expert Commentary
The article presents a novel and innovative approach to data assimilation, which has the potential to revolutionize the field. The proposed PICSB model demonstrates state-of-the-art performance on fluid PDE benchmarks, and its ability to learn without high-fidelity supervision is a significant advancement. However, the model also raises several concerns, including potential overfitting to PDE residuals and increased computational complexity. These limitations should be addressed in future work to ensure the widespread adoption of PICSB in real-world applications.
Recommendations
- ✓ Future work should focus on developing more robust methods for incorporating PDE residuals and hard conditioning to prevent overfitting.
- ✓ The computational complexity of the iterative surrogate-endpoint refresh scheme should be further optimized to enable large-scale applications.
Sources
Original: arXiv - cs.LG
