Emulating Non-Differentiable Metrics via Knowledge-Guided Learning: Introducing the Minkowski Image Loss

2026-04-13

AI Summary

This work addresses the challenge in Earth system deep learning where non-differentiable scientific metrics—such as area, perimeter, and connectivity—cannot be directly optimized, often yielding blurry outputs with lost high-frequency details. To overcome this, the study introduces the first differentiable loss formulation based on Minkowski functionals, achieved through temperature-controlled sigmoid relaxation and continuous logical operators to enforce geometric constraints. Furthermore, it proposes a Lipschitz convolutional network that integrates spectral normalization with hard geometric constraints, enabling high-fidelity surrogate learning of non-differentiable metrics. Evaluated on the EUMETNET OPERA dataset, the method completely eliminates geometric violations and significantly outperforms unconstrained baselines, while also uncovering an inherent trade-off between Lipschitz regularization and local texture recovery.

Technology Category

• Computer Vision: Remote Sensing / Geospatial AI
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The ``differentiability gap'' presents a primary bottleneck in Earth system deep learning: since models cannot be trained directly on non-differentiable scientific metrics and must rely on smooth proxies (e.g., MSE), they often fail to capture high-frequency details, yielding ``blurry'' outputs. We develop a framework that bridges this gap using two different methods to deal with non-differentiable functions: the first is to analytically approximate the original non-differentiable function into a differentiable equivalent one; the second is to learn differentiable surrogates for scientific functionals. We formulate the analytical approximation by relaxing discrete topological operations using temperature-controlled sigmoids and continuous logical operators. Conversely, our neural emulator uses Lipschitz-convolutional neural networks to stabilize gradient learning via: (1) spectral normalization to bound the Lipschitz constant; and (2) hard architectural constraints enforcing geometric principles. We demonstrate this framework's utility by developing the Minkowski image loss, a differentiable equivalent for the integral-geometric measures of surface precipitation fields (area, perimeter, connected components). Validated on the EUMETNET OPERA dataset, our constrained neural surrogate achieves high emulation accuracy, completely eliminating the geometric violations observed in unconstrained baselines. However, applying these differentiable surrogates to a deterministic super-resolution task reveals a fundamental trade-off: while strict Lipschitz regularization ensures optimization stability, it inherently over-smooths gradient signals, restricting the recovery of highly localized convective textures. This work highlights the necessity of coupling such topological constraints with stochastic generative architectures to achieve full morphological realism.

Problem

• differentiability gap
• non-differentiable metrics
• Earth system deep learning
• scientific functionals
• Minkowski image loss

Innovation

• differentiable surrogate
• Minkowski image loss
• Lipschitz-constrained CNN
• topological regularization
• knowledge-guided learning