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REVIEW 3 major objections 7 minor 1 cited by

Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A single metric, EcoL2, turns accuracy and lifecycle carbon emissions into one score, and reorders neural PDE solvers once emissions are counted.

desk verdict The paper's empirical carbon-emissions message is solid, but EcoL2 rankings flip when you change units from kg to grams, so the metric needs major revision before it can be used. read the letter →

arxiv 2505.12556 v1 pith:AXZJX5BM submitted 2025-05-18 cs.LG cs.AI

Taniya Kapoor , Abhishek Chandra , Anastasios Stamou , Stephen J Roberts This is my paper · ORCID
classification cs.LGcs.AI
keywords EcoL2metricneuralPDEsolverscarbonfootprintphysics-informednetworksoperatorslifecycleassessmentsustainablemachinelearningrelativeerror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that evaluating neural PDE solvers by accuracy alone is misleading, because solvers with nearly identical relative error can differ by large factors in carbon emissions. It introduces EcoL2, a score that combines relative L2 error with lifecycle carbon, summing embodied, developmental, operational, and inference emissions, with higher values meaning better combined performance. On canonical PDE benchmarks the score reorders the field: for the advection equation, PINNsFormer and SPINN have comparable relative error but EcoL2 scores of 0.022 versus 0.103, driven mainly by PINNsFormer's higher tuning and training emissions. The paper also shows the metric responds to application priorities through two weighting hyperparameters, and that the same solver's score changes with hardware and with the carbon intensity of the electricity grid where it runs. If the empirical demonstration holds, EcoL2 gives the community a way to compare solvers on both predictive performance and long-term environmental cost.

What carries the argument

The load-bearing object is the EcoL2 ratio itself. Its numerator, $1-e^{\log_\alpha R}=1-R^{1/\ln\alpha}$, maps relative error into $(0,1)$ with diminishing returns as error shrinks; its denominator, $1+\beta(C_e+C_d+C_o+C_i\,n_{\text{infer}})$, scales the full lifecycle carbon bill by $\beta$. The carbon sum is what makes the metric new: it forces data generation, hyperparameter search, final training, and repeated deployment to be counted alongside error. Proofs that the score is bounded and moves in the intended directions, together with the tunable hyperparameters, are what let the metric serve both high-accuracy and low-carbon regimes.

What would settle it

Re-run the six benchmark solvers on the same four machines while recording energy draw with an external power meter, and compute EcoL2 using measured energy and independent regional grid intensities; if the ordering of PINNsFormer versus SPINN on advection or CNO versus DON on Kuramoto-Sivashinsky changes, or if the tracker's kgCO2 values deviate from the meter beyond noise, the paper's central empirical demonstration does not survive.

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Extended reading notes

Core claim

The central discovery is that accuracy and carbon footprint are separable dimensions of a neural PDE solver's worth, and that a single scalar can carry both. EcoL2 is defined as $$\text{EcoL2} = \frac{1 - $e^{{\log_\alpha R}}$}{1 + \$\beta$(C_e + C_d + C_o + C_i \cdot n_{\text{infer}})},$$ where $R$ is relative L2 error, the $C$ terms are embodied, developmental, operational, and inference carbon, and $\alpha,\beta$ weight accuracy versus sustainability. The paper proves the score lies in $(0,1)$, approaches 1 exactly when $R\to 0$ and total carbon $C\to 0$, and approaches 0 when the solver is inaccurate or the carbon bill grows without bound. Across advection, reaction, wave, KdV, and Kuramoto-Sivashinsky equations, the empirical claim is that accuracy-only rankings hide large emission differences, so comparable models can receive very different EcoL2 scores. The paper treats EcoL2 as a general evaluation protocol rather than a PDE-specific benchmark, applying the same score to function approximation and symbolic regression.

Load-bearing premise

Every EcoL2 value inherits the accuracy of the emission estimate $C = P \times t \times I$ that the paper's tracker derives from power draw, runtime, and regional grid carbon intensity; if that estimate is unreliable for a solver or machine, the empirical rankings EcoL2 produces inherit the error.

Editorial extensions

If this is right

  • EcoL2 turns model selection into a two-criterion decision: a solver with slightly worse error can be the better choice if its lifecycle carbon is far lower, and the paper gives specific cases where this flips the ranking.
  • Hyperparameter tuning emerges as a dominant emission term for PINN-family solvers, so reporting EcoL2 pushes developers to count tuning cost rather than only final training error.
  • Because $\alpha$ and $\beta$ are tunable, the same benchmark can rank the best solver differently for accuracy-first applications versus large-scale deployment, making the metric explicitly application-dependent.
  • Hardware and geography enter the score through power draw and grid carbon intensity, so EcoL2 provides a way to compare not only algorithms but also the conditions under which they run.
  • The metric carries over to non-PDE scientific machine-learning tasks such as function approximation and symbolic regression, implying it could serve as a general accuracy-carbon score for scientific models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The denominator's carbon terms are dimensionful, so EcoL2's numerical value depends on the mass unit chosen for CO2; if the field adopts the metric, a standard reporting unit such as kgCO2 will be necessary for scores to be comparable across papers.
  • Because the proof of Lemma 1 requires $R\to 0$ and $C\to 0$ simultaneously, EcoL2 will never reach 1 in practice, so useful thresholds would need to be calibrated on existing benchmarks.
  • The strong dependence on developmental carbon suggests that cheaper tuning strategies, such as shared hyperparameter sweeps or prior-informed search, could improve EcoL2 as much as better architectures; that hypothesis is testable but not established by the paper.
  • Once EcoL2 becomes a standard reporting target, teams that report only error will be visibly hiding carbon, a practical reversal of the paper's own argument that a metric which becomes the target can stop being a good measure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. This paper introduces EcoL2, a scalar metric for neural PDE solvers that combines relative L2 error with lifecycle carbon emissions. The metric is defined in Eq. (2) as [1 - exp(log_alpha R)] / [1 + beta(Ce + Cd + C0 + Ci * n_infer)]. The authors classify emissions into embodied, developmental, operational, and inference carbon; measure them with CodeCarbon; and present experiments on PINN-family solvers and neural operators for the advection, reaction, wave, KdV, and KS equations, as well as on function approximation and symbolic regression. They prove bounds and limit statements for EcoL2 (Proposition 1 and Lemmas 1-2), show that solvers with similar relative error can have very different carbon footprints, and argue that EcoL2 should be used for sustainable model selection.

Significance. If the metric were unit-consistent, the paper would provide a timely and useful benchmark for Green AI in scientific machine learning. The empirical observation that solvers with comparable relative error differ substantially in CodeCarbon-measured emissions is plausible and is the paper's strongest empirical asset. The manuscript ships reproducible pseudo-code, detailed hyperparameter tables, and CodeCarbon tracking scripts, which is a definite strength. The theoretical analysis, however, is largely definitional: the bounds and limits follow immediately from the formula and do not constitute a substantive derivation. The main value of the paper lies in the proposed evaluation lens and the empirical case studies, not in the formal theory.

major comments (3)
  1. [Section 3, Eq. (2); 'Dimensional analysis' paragraph; Proposition 1] The definition of EcoL2 is not dimensionally consistent, and as a result the rankings it produces are not invariant to the arbitrary choice of mass unit. The numerator is dimensionless, while the denominator is 1 + beta*(... kgCO2 ...). For EcoL2 to be a pure number in (0,1), beta must carry units of kgCO2^{-1}; however, the paper treats beta as a dimensionless hyperparameter with beta >= 1 and sets beta = 100, and the 'Dimensional analysis' paragraph states that EcoL2 has units kgCO2^{-1}. These statements contradict Proposition 1. The consequence is concrete: in Table 2, recomputing the same emissions in grams while keeping beta = 100 changes GP from about 0.888 to 0.800 and NP from about 0.881 to 0.853, reversing their ranking. Because the paper's central claim is that higher EcoL2 is preferable, the metric must either specify beta's units explicitly or define C relative to a fixed reference emission so that the score is unit-invariant.
  2. [Section 3, 'Measuring carbon'; Section 4.2] The empirical case for EcoL2 rests on CodeCarbon's emission model C = P * t * I being accurate for all four machines and six countries, but no sensitivity or uncertainty analysis is provided. If CodeCarbon's power-draw estimates or regional carbon intensities are biased, the EcoL2 rankings and the order-of-magnitude differences reported in Tables 1 and 2 could change. The paper should report the raw power, runtime, and intensity values underlying each reported C, and should discuss or bound the sensitivity of the rankings to plausible errors in these quantities.
  3. [Section 3, Eqs. (1)-(2); Algorithm 1; Table 1; Figure 14] The role of n_infer is ambiguous. Equation (1) defines total carbon as Ce + Cd + Co + Ci, while Eq. (2) and Algorithm 1 use Ci * n_infer. Table 1 reports values for Ci and C that are consistent with n_infer = 1, yet Figure 14 is captioned '100 Inferences'. For example, in the KdV/FNO row, Ce + Cd + Co equals the reported C to three significant digits, which is inconsistent with multiplying Ci by 100. The authors should clarify whether Ci denotes per-inference emissions or total inference emissions, and correspondingly whether Eq. (1) should read Ce + Cd + Co + Ci * n_infer.
minor comments (7)
  1. [Eq. (2); Section 4; Tables] The notation for operational carbon is inconsistent: Eq. (2) uses C0, while the text, Table 1, and Table 13 use Co; the notation should be unified.
  2. [Section 3, 'Measuring carbon'] The sentence 'CodeCarbon estimates emissions C2' appears to contain a typo; it should read 'C' rather than 'C2'.
  3. [Table 1; Section E] Table 1 uses the header 'MEA' while the text and Section E define 'MAE'; one spelling should be used throughout.
  4. [Section 4.1; Section 3] Section 4.1 uses beta for the advection wave speed, while Section 3 uses beta as the EcoL2 weighting hyperparameter; the collision could confuse readers and one symbol should be renamed.
  5. [Table 13] The DNN total carbon value is printed as '1.58289' without an exponent; from the preceding columns it should be 1.58289e-6.
  6. [Section H.1; Section 1] There are minor typos: 'ratioanle' in Section H.1 and 'acccuracy' in Section 1.
  7. [Figure 3] Figure 3 would be easier to read if the axes were labeled and the fixed parameters for each panel were stated in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EcoL2 is a proposed definition with analytic properties, and the empirical carbon-footprint comparisons are independent measurements.

full rationale

EcoL2 is introduced in Eq. (2) as a proposed definition, not as the output of a derivation from other premises. Proposition 1, Lemmas 1 and 2, and Corollary 1 restate, in limit or bound form, the algebraic properties of that formula: for R in (0,0.1) the numerator lies in (0,1), and the denominator is strictly greater than 1 under the stated assumptions. These are analytic consequences of the definition rather than empirical predictions, and no parameter is fitted to produce them. The central empirical demonstration, that models with comparable errors can have different carbon footprints, uses CodeCarbon emissions tracked separately at each lifecycle stage; it is not an artifact of the EcoL2 formula's own inputs. No load-bearing uniqueness theorem or prior-work citation is invoked. The paper's earlier works appear only for context (e.g., a beam-dynamics application and sampled neural networks) and for code provenance, and none of those citations is used to justify the metric's validity. The dimensional-analysis paragraph stating that EcoL2 has units kgCO2^-1 is internally inconsistent with the boundedness claim and creates a unit-dependence concern for ranking comparisons, since expressing emissions in grams rather than kilograms can alter scores when beta is treated as dimensionless; however, that is a correctness and robustness issue, not a circularity issue. The score is therefore 0.

Assumptions & free parameters 4 free parameters · 3 assumptions · 1 invented entities

The central claim rests on the measurement model of CodeCarbon, on the assumed additivity of four carbon stages, and on hand-set weights alpha and beta plus the R < 0.1 domain restriction. No physical entities are invented.

free parameters (4)
  • alpha = 100 (main results; varied 10-1000 in ablations)
    Weight on accuracy in the numerator; set by hand and arbitrary; changes the score and can flip model rankings.
  • beta = 100 (main results; varied in ablations)
    Scaling weight on carbon in the denominator; set by hand; controls the accuracy-sustainability tradeoff.
  • n_infer = 1 (main tables); 100 (Figure 14 and pie charts)
    Number of inference copies attributed to deployment; required to compute C_i * n_infer; choice affects total carbon.
  • R accuracy threshold = 0.1
    Definition 1 declares solvers with R >= 0.1 'inaccurate'; restricts the theoretical domain to R in (0, 0.1), though some SM tables include R near 1.
assumptions (3)
  • domain assumption CodeCarbon's emission estimate C = P * t * I represents true per-stage emissions
    Section 3 'Measuring carbon'; all empirical EcoL2 values inherit the accuracy of this model.
  • domain assumption Lifecycle carbon is the disjoint sum C_e + C_d + C_o + C_i * n_infer
    Equation (1); assumes no double counting and complete coverage of the four stages.
  • ad hoc to paper Domain restrictions R in (0, 0.1), alpha in [10, 1000], beta >= 1, C > 0
    Assumptions in Section 3 that make Proposition 1 true; chosen after the fact to force EcoL2 into (0, 1).
invented entities (1)
  • EcoL2 metric
    purpose: Unified ranking score combining relative error and lifecycle carbon emissions
    The paper gives no external criterion showing high EcoL2 correlates with desired outcomes; it is a definition, not a discovered quantity.

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Cite this review

Pith. "Pith review of Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers." pith.science (2026). https://pith.science/paper/AXZJX5BM

@misc{pith2026250512556,
  author       = {Pith},
  title        = {Pith review of: Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AXZJX5BM}},
  note         = {Machine review of arXiv:2505.12556}
}
read the original 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

Figures reproduced from arXiv: 2505.12556 by the authors.

Figure 1
Figure 1. Performance comparison of physics-informed learning methods and neural operators [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Lifecycle carbon of neural PDE solvers The first source is the data required to train these solvers. The carbon associated with this stage is referred to as embodied carbon (Ce) and ac￾counts for emissions associated with data generation. For forward prob￾lems, PINN often requires no training data beyond the well-posed PDE and does not contribute to embodied car￾bon. In contrast, neural operator models rely on large… view at source ↗
Figure 3
Figure 3. EcoL2 for varying α, β values Motivation: The formulation of EcoL2 integrates accu￾racy and environmental cost into a single score. EcoL2’s numerator (hereafter referred to as numerator) uses an exponential-log transformation of the relative L2 error, e logα R, and subtracts it from unity to capture the nonlin￾ear value of accuracy gains. Modulating by α emphasizes that EcoL2 score improvements in low-error regimes … view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: EcoL2’s adaptive performance: KdV (a, b) and KS (c, d) PDEs for varying [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Impact of different machines Impact of hardware choice on emissions: EcoL2 metric also considers the impact of hardware choice by consid￾7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Country-wise variations Cross-regional emissions: EcoL2 metric con￾siders the carbon intensity (I) of the region where the solver is run (presented briefly in SM§I) as illustrated in Section 3. This experi￾ment presents an ablation study for the KS equa￾tion simulated …
Figure 7
Figure 7. Figure 7: Pseudo-code for computing Ce. 1 import numpy as np 2 # Import the emissions tracker from CodeCarbon 3 from codecarbon import EmissionsTracker 4 5 # Initialize the emissions tracker to track developmental carbon 6 tracker = EmissionsTracker ( 7 project_name =" Developme…
Figure 8
Figure 8. Figure 8: Pseudo-code for computing Cd. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Pseudo-code for computing Co. 1 import numpy as np 2 # Import the emissions tracker from CodeCarbon 3 from codecarbon import EmissionsTracker 4 5 # Initialize the emissions tracker to track developmental carbon 6 tracker = EmissionsTracker ( 7 project_name =" Inference…
Figure 10
Figure 10. Figure 10: Pseudo-code for computing Ci . 18 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Pseudo-code for computing EcoL2 score for neural PDE solvers. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Pseudo-code for computing EcoL2 for a deep learning model or a numerical solver. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Rows show performance comparison of physics-informed learning methods on the reaction [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Component-wise Carbon Emissions across PDE Solvers (100 Inferences) [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: Performance comparison of operator learning methods on the KS equation. Models are [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 16
Figure 16. Figure 16: Function approximation performance. Left: GP; Middle: NP; Right: DNN [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]
Figure 17
Figure 17. Figure 17: ODE Discovery [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]

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Reference graph

Works this paper leans on

74 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    Making partial differential equations accessible to ecologists.Nature Reviews Biodiversity, 1(2):87–87, 2025

    Ruby Krasnow. Making partial differential equations accessible to ecologists.Nature Reviews Biodiversity, 1(2):87–87, 2025

    work page 2025

  2. [2]

    Kjetil O Lye, Siddhartha Mishra, Deep Ray, and Praveen Chandrashekar. Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks.Computer Methods in Applied Mechanics and Engineering, 374: 113575, 2021

    work page 2021

  3. [3]

    Neural differential equation-based two-stage approach for generalization of beam dynamics.IEEE Transactions on Industrial Informatics, 2024

    Taniya Kapoor, Hongrui Wang, Anastasios Stamou, Kareem El Sayed, Alfredo Núñez, Daniel M Tartakovsky, and Rolf Dollevoet. Neural differential equation-based two-stage approach for generalization of beam dynamics.IEEE Transactions on Industrial Informatics, 2024

    work page 2024

  4. [4]

    Springer Science & Business Media, 2008

    Alfio Quarteroni and Alberto Valli.Numerical approximation of partial differential equations, volume 23. Springer Science & Business Media, 2008

    work page 2008

  5. [5]

    Cambridge university press, 2002

    Randall J LeVeque.Finite volume methods for hyperbolic problems, volume 31. Cambridge university press, 2002

    2002

  6. [6]

    Partial differential equations meet deep neural networks: A survey.IEEE Transactions on Neural Networks and Learning Systems, 2025

    Shudong Huang, Wentao Feng, Chenwei Tang, Zhenan He, Caiyang Yu, and Jiancheng Lv. Partial differential equations meet deep neural networks: A survey.IEEE Transactions on Neural Networks and Learning Systems, 2025

    work page 2025

  7. [7]

    Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations.Nature Machine Intelligence, 6(10):1256–1269, 2024

    Nick McGreivy and Ammar Hakim. Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations.Nature Machine Intelligence, 6(10):1256–1269, 2024

    2024

  8. [8]

    Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    2021

Show all 74 references
  1. [9]

    Neural operators for accelerating scientific simulations and design

    Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, and Anima Anandkumar. Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, 6(5):320–328, 2024

    2024

  2. [10]

    An operator preconditioning perspective on training in physics-informed machine learning

    Tim De Ryck, Florent Bonnet, Siddhartha Mishra, and Emmanuel de Bezenac. An operator preconditioning perspective on training in physics-informed machine learning. InThe Twelfth International Conference on Learning Representations, 2024

    2024

  3. [11]

    Achieving high accuracy with PINNs via energy natural gradient descent

    Johannes Müller and Marius Zeinhofer. Achieving high accuracy with PINNs via energy natural gradient descent. InInternational Conference on Machine Learning, pages 25471–25485. PMLR, 2023

    2023

  4. [12]

    Solving partial differential equations with sampled neural networks.arXiv preprint arXiv:2405.20836, 2024

    Chinmay Datar, Taniya Kapoor, Abhishek Chandra, Qing Sun, Iryna Burak, Erik Lien Bolager, Anna Veselovska, Massimo Fornasier, and Felix Dietrich. Solving partial differential equations with sampled neural networks.arXiv preprint arXiv:2405.20836, 2024

    2024 arXiv

  5. [13]

    Reliance on metrics is a fundamental challenge for AI

    Rachel L Thomas and David Uminsky. Reliance on metrics is a fundamental challenge for AI. Patterns, 3(5), 2022

    2022

  6. [14]

    Problems of monetary management: the uk experience in papers in monetary economics.Monetary Economics, 1, 1975

    Charles Goodhart. Problems of monetary management: the uk experience in papers in monetary economics.Monetary Economics, 1, 1975

    1975

  7. [15]

    M Raissi, P Perdikaris, and GE Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019. 10

    2019

  8. [16]

    Pinnsformer: A transformer-based framework for physics-informed neural networks

    Zhiyuan Zhao, Xueying Ding, and B Aditya Prakash. Pinnsformer: A transformer-based framework for physics-informed neural networks. InThe Twelfth International Conference on Learning Representations, 2024

    2024

  9. [17]

    Separable physics-informed neural networks.Advances in Neural Information Processing Systems, 36:23761–23788, 2023

    Junwoo Cho, Seungtae Nam, Hyunmo Yang, Seok-Bae Yun, Youngjoon Hong, and Eunbyung Park. Separable physics-informed neural networks.Advances in Neural Information Processing Systems, 36:23761–23788, 2023

    2023

  10. [18]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218–229, 2021

    2021

  11. [19]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar, et al. Fourier neural operator for parametric partial differential equations. InInternational Conference on Learning Representations, 2021

    2021

  12. [20]

    Convolutional neural operators for robust and accurate learning of PDEs.Advances in Neural Information Processing Systems, 36: 77187–77200, 2023

    Bogdan Raonic, Roberto Molinaro, Tim De Ryck, Tobias Rohner, Francesca Bartolucci, Rima Alaifari, Siddhartha Mishra, and Emmanuel de Bézenac. Convolutional neural operators for robust and accurate learning of PDEs.Advances in Neural Information Processing Systems, 36: 77187–77...

    2023

  13. [21]

    PdeBench: An extensive benchmark for scientific machine learning.Advances in Neural Information Processing Systems, 35:1596–1611, 2022

    Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Daniel MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert. PdeBench: An extensive benchmark for scientific machine learning.Advances in Neural Information Processing Systems, 35:1596–1611, 2022

    2022

  14. [22]

    The well: a large-scale collection of diverse physics simulations for machine learning.Advances in Neural Information Processing Systems, 37:44989–45037, 2024

    Ruben Ohana, Michael McCabe, Lucas Meyer, Rudy Morel, Fruzsina Agocs, Miguel Beneitez, Marsha Berger, Blakesly Burkhart, Stuart Dalziel, Drummond Fielding, et al. The well: a large-scale collection of diverse physics simulations for machine learning.Advances in Neural Informat...

    2024

  15. [23]

    Automating embodied and operational carbon assessment in urban sustainable development.Automation in Construction, 176:106245, 2025

    Siavash Ghorbany, Ming Hu, Siyuan Yao, Matthew Sisk, and Chaoli Wang. Automating embodied and operational carbon assessment in urban sustainable development.Automation in Construction, 176:106245, 2025

    2025

  16. [24]

    Estimates on the generalization error of physics- informed neural networks for approximating PDEs.IMA Journal of Numerical Analysis, 43(1): 1–43, 2023

    Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics- informed neural networks for approximating PDEs.IMA Journal of Numerical Analysis, 43(1): 1–43, 2023

    2023

  17. [25]

    Poseidon: Efficient foundation models for PDEs

    Maximilian Herde, Bogdan Raonic, Tobias Rohner, Roger Käppeli, Roberto Molinaro, Em- manuel de Bézenac, and Siddhartha Mishra. Poseidon: Efficient foundation models for PDEs. Advances in Neural Information Processing Systems, 37:72525–72624, 2024

    2024

  18. [26]

    Gaussian processes for machine learning.International journal of neural systems, 14(02):69–106, 2004

    Matthias Seeger. Gaussian processes for machine learning.International journal of neural systems, 14(02):69–106, 2004

    2004

  19. [27]

    Neural processes.arXiv preprint arXiv:1807.01622, 2018

    Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J Rezende, SM Eslami, and Yee Whye Teh. Neural processes.arXiv preprint arXiv:1807.01622, 2018

    2018 arXiv

  20. [28]

    Deep learning.nature, 521(7553):436–444, 2015

    Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning.nature, 521(7553):436–444, 2015

    2015

  21. [29]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the national academy of sciences, 113(15):3932–3937, 2016

    Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the national academy of sciences, 113(15):3932–3937, 2016

    2016

  22. [30]

    Tackling the curse of dimensionality with physics-informed neural networks.Neural Networks, 176:106369, 2024

    Zheyuan Hu, Khemraj Shukla, George Em Karniadakis, and Kenji Kawaguchi. Tackling the curse of dimensionality with physics-informed neural networks.Neural Networks, 176:106369, 2024

    2024

  23. [31]

    Physics informed neural networks for simulating radiative transfer.Journal of Quantitative Spectroscopy and Radiative Transfer, 270:107705, 2021

    Siddhartha Mishra and Roberto Molinaro. Physics informed neural networks for simulating radiative transfer.Journal of Quantitative Spectroscopy and Radiative Transfer, 270:107705, 2021. 11

    2021

  24. [32]

    On the morality of artificial intelligence [commentary]

    Alexandra Luccioni and Yoshua Bengio. On the morality of artificial intelligence [commentary]. IEEE Technology and Society Magazine, 39(1):16–25, 2020

    2020

  25. [33]

    Measuring the carbon intensity of AI in cloud instances

    Jesse Dodge, Taylor Prewitt, Remi Tachet des Combes, Erika Odmark, Roy Schwartz, Emma Strubell, Alexandra Sasha Luccioni, Noah A Smith, Nicole DeCario, and Will Buchanan. Measuring the carbon intensity of AI in cloud instances. InProceedings of the 2022 ACM conference on fairn...

    2022

  26. [34]

    Quantifying the carbon emissions of machine learning.arXiv preprint arXiv:1910.09700, 2019

    Alexandre Lacoste, Alexandra Luccioni, Victor Schmidt, and Thomas Dandres. Quantifying the carbon emissions of machine learning.arXiv preprint arXiv:1910.09700, 2019

    1910 arXiv

  27. [35]

    Green AI.Communications of the ACM, 63(12):54–63, 2020

    Roy Schwartz, Jesse Dodge, Noah A Smith, and Oren Etzioni. Green AI.Communications of the ACM, 63(12):54–63, 2020

    2020

  28. [36]

    Energy and carbon considerations of fine-tuning BERT

    Xiaorong Wang, Clara Na, Emma Strubell, Sorelle Friedler, and Sasha Luccioni. Energy and carbon considerations of fine-tuning BERT. InFindings of the Association for Computational Linguistics: EMNLP 2023, pages 9058–9069, 2023

    2023

  29. [37]

    Bloom: A 176B-parameter open-access multilingual language model, 2023

    Teven Le Scao, Angela Fan, Christopher Akiki, Ellie Pavlick, Suzana Ili ´c, Daniel Hesslow, Roman Castagné, Alexandra Sasha Luccioni, François Yvon, Matthias Gallé, et al. Bloom: A 176B-parameter open-access multilingual language model, 2023

    2023

  30. [38]

    Estimating the carbon footprint of BLOOM, a 176B parameter language model.Journal of Machine Learning Research, 24(253):1–15, 2023

    Alexandra Sasha Luccioni, Sylvain Viguier, and Anne-Laure Ligozat. Estimating the carbon footprint of BLOOM, a 176B parameter language model.Journal of Machine Learning Research, 24(253):1–15, 2023

    2023

  31. [39]

    GPT-4 technical report.arXiv preprint arXiv:2303.08774, 2023

    Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. GPT-4 technical report.arXiv preprint arXiv:2303.08774, 2023

    2023 arXiv

  32. [40]

    https://news.mit.edu/ 2025/explained-generative-ai-environmental-impact-0117, 2025

    Explained: Generative AI’s environmental impact — news.mit.edu. https://news.mit.edu/ 2025/explained-generative-ai-environmental-impact-0117, 2025

    2025

  33. [41]

    The carbon impact of artificial intelligence.Nature Machine Intelligence, 2 (8):423–425, Aug 2020

    Payal Dhar. The carbon impact of artificial intelligence.Nature Machine Intelligence, 2 (8):423–425, Aug 2020. ISSN 2522-5839. doi: 10.1038/s42256-020-0219-9. URL https: //doi.org/10.1038/s42256-020-0219-9

    2020 doi

  34. [42]

    A high-efficient hybrid physics-informed neural networks based on convolutional neural network.IEEE Transactions on Neural Networks and Learning Systems, 33(10):5514– 5526, 2021

    Zhiwei Fang. A high-efficient hybrid physics-informed neural networks based on convolutional neural network.IEEE Transactions on Neural Networks and Learning Systems, 33(10):5514– 5526, 2021

    2021

  35. [43]

    Gnot: A general neural operator transformer for operator learning

    Zhongkai Hao, Zhengyi Wang, Hang Su, Chengyang Ying, Yinpeng Dong, Songming Liu, Ze Cheng, Jian Song, and Jun Zhu. Gnot: A general neural operator transformer for operator learning. InInternational Conference on Machine Learning, pages 12556–12569. PMLR, 2023

    2023

  36. [44]

    In-context operator learning with data prompts for differential equation problems.Proceedings of the National Academy of Sciences, 120(39):e2310142120, 2023

    Liu Yang, Siting Liu, Tingwei Meng, and Stanley J Osher. In-context operator learning with data prompts for differential equation problems.Proceedings of the National Academy of Sciences, 120(39):e2310142120, 2023

    2023

  37. [45]

    Alias-free Mamba neural operator.Advances in Neural Information Processing Systems, 37:52962–52995, 2024

    Jianwei Zheng, Wei Li, Ni Xu, Junwei Zhu, and Xiaoqin Zhang. Alias-free Mamba neural operator.Advances in Neural Information Processing Systems, 37:52962–52995, 2024

    2024

  38. [46]

    Codecarbon: estimate and track carbon emissions from machine learning computing.Cited on, 20, 2021

    Victor Schmidt, Kamal Goyal, Aditya Joshi, Boris Feld, Liam Conell, Nikolas Laskaris, Doug Blank, Jonathan Wilson, Sorelle Friedler, and Sasha Luccioni. Codecarbon: estimate and track carbon emissions from machine learning computing.Cited on, 20, 2021

    2021

  39. [47]

    Carbontracker: Track- ing and predicting the carbon footprint of training deep learning models.arXiv preprint arXiv:2007.03051, 2020

    Lasse F Wolff Anthony, Benjamin Kanding, and Raghavendra Selvan. Carbontracker: Track- ing and predicting the carbon footprint of training deep learning models.arXiv preprint arXiv:2007.03051, 2020

    2007 arXiv

  40. [48]

    Towards the systematic reporting of the energy and carbon footprints of machine learning

    Peter Henderson, Jieru Hu, Joshua Romoff, Emma Brunskill, Dan Jurafsky, and Joelle Pineau. Towards the systematic reporting of the energy and carbon footprints of machine learning. Journal of Machine Learning Research, 21(248):1–43, 2020. 12

    2020

  41. [49]

    Eco2ai: carbon emissions tracking of machine learning models as the first step towards sustainable ai

    Semen Andreevich Budennyy, Vladimir Dmitrievich Lazarev, Nikita Nikolaevich Zakharenko, Aleksei N Korovin, OA Plosskaya, Denis Valer’evich Dimitrov, VS Akhripkin, IV Pavlov, Ivan Valer’evich Oseledets, Ivan Segundovich Barsola, et al. Eco2ai: carbon emissions tracking of machi...

    2022

  42. [50]

    Green algorithms: quantifying the carbon footprint of computation.Advanced science, 8(12):2100707, 2021

    Loïc Lannelongue, Jason Grealey, and Michael Inouye. Green algorithms: quantifying the carbon footprint of computation.Advanced science, 8(12):2100707, 2021

    2021

  43. [51]

    Neural-network-based approximations for solving partial differential equations.communications in Numerical Methods in Engineering, 10(3):195–201, 1994

    MWM Gamini Dissanayake and Nhan Phan-Thien. Neural-network-based approximations for solving partial differential equations.communications in Numerical Methods in Engineering, 10(3):195–201, 1994

    1994

  44. [52]

    Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5): 987–1000, 1998

    Isaac E Lagaris, Aristidis Likas, and Dimitrios I Fotiadis. Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5): 987–1000, 1998

    1998

  45. [53]

    Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.Science advances, 7(40): eabi8605, 2021

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.Science advances, 7(40): eabi8605, 2021

    2021

  46. [54]

    Physics-informed deep neural operator networks

    Somdatta Goswami, Aniruddha Bora, Yue Yu, and George Em Karniadakis. Physics-informed deep neural operator networks. InMachine learning in modeling and simulation: methods and applications, pages 219–254. Springer, 2023

    2023

  47. [55]

    https://github.com/ taniyakapoor/PINNs_beam/tree/master, 2023

    GitHub - taniyakapoor/PINNs_beam at master — github.com. https://github.com/ taniyakapoor/PINNs_beam/tree/master, 2023. [Accessed 18-05-2025]

    2023

  48. [56]

    https://github.com/AdityaLab/ pinnsformer, 2024

    GitHub - AdityaLab/pinnsformer — github.com. https://github.com/AdityaLab/ pinnsformer, 2024. [Accessed 18-05-2025]

    2024

  49. [57]

    https://github.com/stnamjef/SPINN, 2023

    GitHub - stnamjef/SPINN: [NeurIPS 2023 Spotlight] Separable Physics-Informed Neural Networks — github.com. https://github.com/stnamjef/SPINN, 2023. [Accessed 18-05- 2025]

    2023

  50. [58]

    https://github.com/raj-brown/ APMA2070, 2020

    GitHub - raj-brown/APMA2070 — github.com. https://github.com/raj-brown/ APMA2070, 2020. [Accessed 18-05-2025]

    2020

  51. [59]

    — github.com

    GitHub - neuraloperator/neuraloperator: Learning in infinite dimension with neural operators. — github.com. https://github.com/neuraloperator/neuraloperator, 2021. [Accessed 18-05-2025]

    2021

  52. [60]

    2023. GitHub - camlab-ethz/ConvolutionalNeuralOperator: This repository is the official imple- mentation of the paper Convolutional Neural Operators for robust and accurate learning of PDEs — github.com. https://github.com/camlab-ethz/ConvolutionalNeuralOperator,

    2023

  53. [61]

    Automatic differentiation in pytorch

    Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. InNIPS-W, 2017

    2017

  54. [62]

    JAX: composable transformations of Python+NumPy programs, 2018

    James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. URL http://github.com/jax-ml/jax

    2018

  55. [63]

    Matplotlib: A 2d graphics environment computing in science & engineering 9 (3), 90–95.Publisher: IEEE COMPUTER SOC, 2007

    JD Hunter. Matplotlib: A 2d graphics environment computing in science & engineering 9 (3), 90–95.Publisher: IEEE COMPUTER SOC, 2007

    2007

  56. [64]

    Harris, K

    Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Vir- tanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Hal- dane, Jaim...

    2020

  57. [65]

    pandas-dev/pandas: Pandas, February 2020

    The pandas development team. pandas-dev/pandas: Pandas, February 2020. URL https: //doi.org/10.5281/zenodo.3509134

    2020 doi

  58. [66]

    Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J

    Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, E...

    2020

  59. [67]

    Lie point symmetry data augmenta- tion for neural pde solvers.arXiv preprint arXiv:2202.07643, 2022

    Johannes Brandstetter, Max Welling, and Daniel E Worrall. Lie point symmetry data augmenta- tion for neural pde solvers.arXiv preprint arXiv:2202.07643, 2022

    2022 arXiv

  60. [68]

    Lie Point Symmetry Data Augmentation for Neural PDE Solvers

    GitHub - brandstetter-johannes/LPSDA: Repo to the paper "Lie Point Symmetry Data Augmentation for Neural PDE Solvers" — github.com. https://github.com/brandstetter-johannes/LPSDA# produce-datasets-for-kuramoto-shivashinsky-ks-equation , 2023. [Accessed 18-05-2025]

    2023

  61. [69]

    Understanding the difficulty of training deep feedfor- ward neural networks

    Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedfor- ward neural networks. InProceedings of the thirteenth international conference on artificial intelligence and statistics, pages 249–256. JMLR Workshop and Conference Proceedings, 2010

    2010

  62. [70]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. URL https://arxiv.org/abs/1412.6980

    2017 arXiv

  63. [71]

    [Accessed 18-05-2025]

    Labs — gpss.cc.https://gpss.cc/gpss24/labs, 2024. [Accessed 18-05-2025]

    2024

  64. [72]

    Embodied Carbon

    Brian de Silva, Kathleen Champion, Markus Quade, Jean-Christophe Loiseau, J Kutz, and Steven Brunton. Pysindy: A python package for the sparse identification of nonlinear dynamical systems from data.Journal of Open Source Software, 5(49):2104, 2020. 14 A Organization of supple...

    2020

  65. [128]

    Xavier uniform initialization [69] is employed with a fixed seed for reproducibility

    The network used the Tanh activation function. Xavier uniform initialization [69] is employed with a fixed seed for reproducibility. The optimizer is LBFGS, configured with a learning rate of 0.1, 21 maximum evaluation steps of 50,000, and a stopping criterion based on a 1×10 ...

  66. [2023]

    [Accessed 18-05-2025]

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.