Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers

TL;DR

This work introduces EcoL2, a carbon-aware evaluation metric for neural PDE solvers that balances predictive accuracy with lifecycle carbon emissions across data collection, development, training, and deployment. By decomposing emissions into embodied, developmental, operational, and inference components, EcoL2 provides a tunable score that rewards both low relative error $\mathcal{R}$ and low carbon footprint $C$, via $\text{EcoL2} = \frac{1 - e^{\log_{\alpha}(\mathcal{R})}}{1 + \beta (C_e + C_d + C_o + C_i \cdot n_{\text{infer}})}$. The metric is demonstrated on physics-informed neural networks and neural operators across multiple canonical PDEs, revealing that hyperparameter tuning and hardware choices often drive emissions more than accuracy alone. EcoL2 enables application-aware model selection by adjusting $\alpha$ and $\beta$, and highlights the significance of regional carbon intensity and data-generation costs. Overall, EcoL2 advances sustainable scientific machine learning by integrating environmental impact into model evaluation and guiding low-carbon solver development.

Abstract

Real-world systems, from aerospace to railway engineering, are modeled with partial differential equations (PDEs) describing the physics of the system. Estimating robust solutions for such problems is essential. Deep learning-based architectures, such as neural PDE solvers, have recently gained traction as a reliable solution method. The current state of development of these approaches, however, primarily focuses on improving accuracy. The environmental impact of excessive computation, leading to increased carbon emissions, has largely been overlooked. This paper introduces a carbon emission measure for a range of PDE solvers. Our proposed metric, EcoL2, balances model accuracy with emissions across data collection, model training, and deployment. Experiments across both physics-informed machine learning and operator learning architectures demonstrate that the proposed metric presents a holistic assessment of model performance and emission cost. As such solvers grow in scale and deployment, EcoL2 represents a step toward building performant scientific machine learning systems with lower long-term environmental impact.

Figures (18)

• Figure 1: Performance comparison of physics-informed learning methods and neural operators on the advection and KdV equations, respectively. Models are evaluated on relative error (Left), the corresponding carbon emissions (kgCO$_2$) (Middle), and the proposed EcoL2 metric (Right). Comparison of models solely based on relative error provides a myopic view as they have varying carbon footprints. The proposed EcoL2 metric (higher values are preferable) captures this trade-off, offering a performant perspective of solver performance.
• Figure 2: Lifecycle carbon of neural PDE solvers
• Figure 3: EcoL2 for varying $\alpha$, $\beta$ values
• Figure 4: EcoL2's adaptive performance: KdV (a, b) and KS (c, d) PDEs for varying $\alpha$ and $\beta$
• Figure 5: Impact of different machines

Theorems & Definitions (8)

• Definition 1
• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1