PhysE-Inv: A Physics-Encoded Inverse Modeling approach for Arctic Snow Depth Prediction

Technical Deep Dive: PhysE-Inv - Physics-Encoded Inverse Modeling for Arctic Snow Depth Prediction

Overview

PhysE-Inv is a novel framework that integrates a sophisticated sequential architecture, namely an LSTM Encoder-Decoder with Multi-head Attention and contrastive learning, with physics-guided inverse modeling. The core innovation lies in a physics-constrained inversion methodology that leverages the hydrostatic balance forward model as a target-formulation proxy, enabling effective learning in the absence of direct ground truth, and uses reconstruction physics regularization over a latent space to dynamically discover hidden physical parameters from noisy, incomplete time-series input.

Problem & Context

• Accurate estimation of Arctic snow depth remains a critical time-varying inverse problem due to the scarcity in associated sea ice parameters.
• Existing process-based and data-driven models are either highly sensitive to sparse data or lack the physical interpretability required for climate-critical applications.
• The fundamental difficulty stems from the non-unique, ill-posed nature of the inverse mapping between observed variables (e.g., snow density) and the target quantities (e.g., snow depth).

Methodology

Key Components

1. Surjective Inverse Mapping: Addresses the inherent non-uniqueness by defining a surjective mapping that links the model's latent state to the final physical parameters, ensuring dynamic enforcement of physical constraints.
2. Physics Encoding: Leverages the hydrostatic balance equation as a physics-constrained reconstruction proxy to enforce physically consistent predictions of snow depth.
3. Physics-Guided Contrastive Learning (PGCL): Encourages the latent space to capture physically meaningful relationships between Arctic snow and sea ice variables by enforcing these relationships through contrastive loss.

Model Architecture

• Encoder-Decoder LSTM with Multi-head Attention to capture long-range temporal dependencies
• Surjective Inverse Mapping from latent space to physically constrained parameters
• Physics Encoding layer that uses the estimated parameters to enforce hydrostatic balance
• PGCL objective to ensure latent space respects physical invariance

Data & Experimental Setup

• Dataset: ERA5 atmospheric reanalysis data for the central Arctic Ocean (70°N to 85°N) from 2019 to 2023
• Input features: Snow density, snow albedo, sea ice concentration
• Target: Hydrostatic balance-derived proxy for snow depth
• Evaluation metrics: MSE, RMSE

Results

• PhysE-Inv outperformed multiple baseline models, including LSTM, BiLSTM, NeuralODE, and ResNet50, in both base and parameter estimation settings.
• The proposed model demonstrated superior performance in capturing the seasonal dynamics and extreme events in the snow depth proxy time series.
• Ablation studies showed the importance of the physics-guided contrastive learning component, which improved performance across varying data availability scenarios.

Interpretation

• Embedding physical constraints and leveraging physics-guided contrastive learning are crucial for stable recovery of latent physical parameters under data scarcity.
• The surjective inverse mapping approach effectively addresses the inherent non-uniqueness and ill-posedness of the inverse problem.
• PhysE-Inv's ability to capture the full statistical distribution of snow depth anomalies, rather than just point estimates, is valuable for climate applications.

Limitations & Uncertainties

• The study is limited to the central Arctic region and a 5-year period, so the generalizability to other regions and longer timescales is unclear.
• The accuracy of the hydrostatic balance proxy model and its assumptions introduce potential sources of uncertainty.
• The impact of measurement errors and biases in the input features (e.g., snow density) on the model's performance is not fully explored.

What Comes Next

• Incorporate Bayesian uncertainty quantification to validate learned parameters against independent remote-sensing and in-situ data sources.
• Expand the study to other regions and longer time periods to assess the broader applicability of the PhysE-Inv framework.
• Investigate the integration of additional physical constraints and the potential for transfer learning to further improve performance in data-scarce domains.