| 22 Jul 2026
Physics-Based Machine Learning: Opportunities and Challenges for Mantle Convection
Abstract. Geodynamic research revolves around understanding the Earth's subsurface, which often requires solving complex partial differential equations to produce numerical models. Using state-of-the-art solvers to tackle these high-dimensional problems can be computationally demanding, or even prohibitive. A potential solution is to use surrogate models–methods that reduce the dimensionality of the problem while maintaining its general characteristics. In this work, we focus on developing reliable and efficient surrogate models via physics-based machine learning techniques for mantle convection applications, thus mitigating the computational challenges inherent in direct forward modeling of mantle convection. For this purpose, we employ the non-intrusive reduced basis method (NI-RB), which maintains the accuracy of traditional simulations while significantly reducing computational complexity. The performance of these models is compared against high-dimensional finite element mantle convection models generated from ASPECT, an open-source geodynamical simulation software. We show that the adopted approach significantly speeds up simulations of mantle temperature distribution-enabling rapid multi-query analyses such as global sensitivity studies and uncertainty quantification-while overcoming typical model reduction issues in geodynamics. The results indicate that our surrogate models not only capture the essential dynamics of mantle convection but also offer a balance of computational efficiency and quality of the model. With the potential to transform how geodynamic modeling studies are conducted, these surrogate models hold promise for a more efficient and effective exploration of Earth's subsurface processes in future studies.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Solid Earth.
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Review for manuscript “Physics-Based Machine Learning: Opportunities and Challenges for Mantle Convection” by Marilina Valatsou and collaborators.
The manuscript presents a set of 2-D mantle convection models performed with ASPECT and then utilized to train Physics-Based Machine Learning models. ASPECT models are kept relatively simple with a constant viscosity within a model but varied between the experiments from 1e22 Pa s to 1e23 Pa s corresponding to Rayleigh number between 1e6 and 1e7. These models, ran for 1 Gyr, show a variability from a few straight plumes to more vigorous convection with more complex shapes plumes. Surrogate models are trained from 100 ASPECT simulations and validated against 10 simulations that are not included in the training set.
Then, reduce order models are performed and compared with the high-fidelity ASPECT solutions. Results show that the surrogate models perform relatively well for viscosity greater than 3e22 Pa s while more discrepancies can emerge for lower viscosities where the structure is more complex.
I think the study is interesting, well-illustrated and the results are encouraging.
Nevertheless, I think that there are a few points that could deserve attention. First, the results are sorted into 3 regimes, called regime 1, regime 2 and regime 3. These are selected “visually”. Although I don’t have problem with visual interpretation, I think that the lack of quantification here is at the origin of potential misunderstanding at several places in the manuscript. Section 3.2 is rather difficult to understand, and the relationship between regimes and entropy is not clear to me, at least not entirely. I think that these sections could deserve reworking. I suggest to introduce the content of section 4.1 earlier and split between methods and results instead of discussion.
Second, I have 2 questions that I think could be addressed in more detail in the manuscript because they are at the core philosophy of the approach:
Or should they target a pointwise fidelity solution? I think this question is particularly important because it conditions how to evaluate errors and what to look for in the models to determine if they are valid or not.
Third, I think it would be worth citing more reduce order modelling approaches, especially applied to geosciences. I know there is some work recently published using these methods, citing them would strengthen the manuscript and bring more visibility to this work and the cited work.
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Across the manuscript some acronyms starting with "M" or "N" are preceded by "an" and sometimes by "a". I don't know what the rule in English is but I suppose that it should be consistent across the manuscript.
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Point-by-point comments:
l.20: "engineering constraints"
In a sense accessing to deep mantle under tens of GPa is a challenge engineers may dream about, but I think you could just state:
"The physical inaccessibility of the deeper layers of the Earth due to extreme conditions..."
l.25-27: I think that adding a reference (or several if possible) about the statement:
"surrogate models which, if constructed proficiently, can achieve similar levels of accuracy as the forward models"
would strengthen it.
Also is it true for any type of model e.g., non-linear, 3-D, coupled with multiphysics, or are there limitations?
l.40-41: "data-driven methods arise from their inability to produce explainable and interpretable models in order to understand the governing subsurface processes"
I don't know what is behind the term "data-driven methods", does it include inverse modelling? But I think this statement is a bit strong, forward and inverse models fulfil different objectives, but saying that data-driven method cannot produce interpretable models is a strong statement that could be very unpopular and wrong, at least for some type of models.
l.82-83: "The general idea of the RB method is to represent the solution as a combination of basis functions and coefficients, making it conceptually similar to a traditional Fourier transform approach"
This is similar to a lot of approaches using function spaces, including Fourier transform but also interpolation for exemple.
l.83-87: Is Fourier transform involved in the method used here?
I understand that it is not, therefore I find this explanation around the Fourier Transform a bit confusing. I understand that this is an attempt to explain the concept behind the approach, but I think that it is not necessary. The paragraph that follows is understandable by itself.
L.102-109: Tolerance seems very important. Yet it is specified "user defined". Are there methods to, if not find the optimal tolerance, at least have an approximation of what is an appropriate tolerance for a given problem?
l.190: "conventionally established" but only one article is referenced. Adding more references would strengthen this statement.
L.227: “which matches the user-defined POD tolerance of 1e−1”
It is not clear to me what are the relationships between a tolerance of 1e-1 and a global mean squared error of 2.83e-3. Would it be possible to briefly elaborate on the link between the two values?
l.271: I think it would be better to replace the word “believe” by another word
l.292-335: about the different regimes. Something is not clear to me.
Regime 1: “Among the training realizations, 34 out of 100 display substantial visual similarity with this regime”
Regime 2: “From the training data set, only 24 out of 100 realizations visually align with Regime 2”
Regime 3: “Notably, 40 out of 100 training realizations correspond visually to this regime”
I do not understand what should be retained from these numbers. Would it be possible to clarify what it means? Is this distribution expected? Is it surprising?
Does “visual” correspondence mean that they were categorized by hand after looking at them, or is there a more quantitative variable (or an ensemble of variables) that help categorizing the models?
What about the 2 remaining realizations? (34+24+40 = 98)
This section about the regimes is confusing to me. Information between the type of models, the errors, their number and their description is mixed, and I admit that I have trouble capturing and understanding the message here. I think that this section would benefit from reworking to make it clearer.
l.346-347: “A total of 110 ASPECT forward simulations were executed to generate the results presented in this work: a training data set consisting of 100 simulations and a validation data set consisting of 10 simulations.”
This is a very clear sentence that gives clear information. I suggest putting it in section 2.2 where the same information is provided indirectly between lines 186 and 193. Of course, I do not suggest removing the information provided in section 2.2 about the methods, but opening the paragraph with this sentence would give the reader a better context.
l.412-416: In my opinion, this part is a bit tricky. Because stated like that, one could understand that the variability observed in ASPECT geodynamic models results from accumulated numerical errors and not from viscosity variation. This would be a terrible consequence, because it would mean that the results of the ASPECT model cannot be interpreted as physically consistent. What happens if you run the same model (exact same initial and boundary conditions and exact same parameters) with a given relatively low viscosity but for various linear solver tolerances? Does it end in relatively similar results below a given tolerance, or does it keep showing variability? Are the model results statistically similar? What is the linear solver tolerance employed for these models?
I think this statement should be made very carefully because it can imply a lot of things.
l.417 & l.419: Typo on viscosity values, a “10” is missing in front of the exponent.
About section 4.1. To me, this section should not be in the discussion, it should be split into two parts. One part should be in section 2, the methods section, and the other part should be in section 3, the results section. Then a discussion about the relationships between the regimes and the entropy can be made in section 4.
Eq.1: grad T is a vector, therefore, (grad T)^2 is an ambiguous notation even if it is very likely that it is its norm squared, I suggest writing it clearly to remove any possible ambiguity.
Eq.2: \dot \varepsilon is indicated as the effective strain-rate, but the strain-rate is a rank-2 tensor, is it its second invariant? Or is the square indicating like in Eq.1 that it is its norm squared?
l.474-475: “The bulk heat flux between the model’s upper and lower boundaries is denoted as q.”
Would it be possible to write the expression of q? Because there are a few points that need clarification:
L.483-501: I am a bit puzzled by the rationale to link regimes, entropy and viscosity. For instance, why the red points located around (2e22 , 0.18) are in regime 1 and not regime 2? Based on entropy they should belong to the same regime don’t they?
To me it looks like the lack of a quantitative basis to categorize models is at the origin of the lack of clarity of the analysis in section 3 and here. I think it would be a great improvement to design a method based on quantitative parameters/variables to categorize models.
l.516: “we believe”
I think that the term “believe” should be avoided.
l.563-568: This point is actually very important. Is the purpose of surrogate models to exactly reproduce high-fidelity solutions, or should it be considered as a statistical approach?
For example, if plumes are not at the same location but their number, their size and their temperature are the same than in the high-fidelity model, a local error evaluation would state that the surrogate is wrong while the physics would be captured. Therefore, it asks the question of error evaluation. Should errors be evaluated using pointwise differences locally and integrated (l2-norm, l-inf norm, etc.) or should there be other metrics to use? This is especially important for systems like mantle convection where, as stated in the discussion, small variations in initial conditions can lead to different results because these systems are very dynamic.
Fig. 7: would an x-axis in log-scale be more appropriate? Viscosity is varied from 1e22 to 1e23 and the curve seem to represent a decreasing exponential. Log-scale may highlight a linear relationship in log-space and an easier reading of the results.