the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Oct 2023
Solving crustal heat transfer for thermochronology using physics-informed neural networks
Shengze Cai
Jean Braun
Abstract. We present a deep learning approach based on the physics-informed neural networks (PINNs) for estimating thermal evolution of the crust during tectonic uplift with a changing landscape. The approach approximates the temperature field of the crust with a deep neural network, which is trained by optimizing the heat advection-diffusion equation under boundary conditions such as initial and final thermal structure, topographic history, and surface and basal temperatures. From the trained neural network of temperature field and the prescribed velocity field, one can predict the temperature history of a given rock particle that can be used to compute the cooling ages of thermochronology. For the inverse problem, the forward model can be combined with a global optimization algorithm that minimizes the misfit between predicted and observed thermochronological data, in order to constrain unknown parameters in the uplift history or boundary conditions. We demonstrate the approach with solutions of one- and three-dimensional forward and inverse models of the crustal thermal evolution, which are consistent with results of the finite-element method. The three-dimensional model simulates the post-orogenic topographic decay of the Dabie Shan, China, with constraints from fission-track and (U-Th)/He ages.
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Ruohong Jiao et al.
Status: open (until 21 Dec 2023)
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RC1: 'Comment on gchron-2023-24', Kendra Murray, 23 Nov 2023
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General Comments:
This manuscript presents a new approach for solving the crustal heat transfer problem in 1D and 3D using a recently developed deep-learning approach called physics-informed neural networks (PINNs). A PINN method can solve PDEs like the advection-diffusion equation with complex boundary conditions, especially as inverse problems, and it is therefore an attractive tool for calculating the thermal structure of the crust given changing surface elevation (1D) or topography (3D)—a common task in the interpretation of cooling ages. This manuscript presents forward and inverse solutions with a simple synthetic scenario in 1D, demonstrating that PINNs can predict crustal thermal profiles and time-temperature paths (forward) similarly to a more traditional numerical finite-element method and resolve the “true” exhumation histories, temperature histories, and synthetic cooling ages well (inverse). These 1D solutions are useful for demonstrating some of the model design choices that control optimization, search algorithms, etc. Then, this contribution moves into 3D, using an example of the Cretaceous-Recent landscape evolution of the Dabie Shan (the site of previous thermochronology studies and a published dataset). Forward and inverse models include changes to both topographic relief and rock uplift (and thus exhumation) rates over time. Inversion results are compared to PECUBE (an existing 3D finite-element method that is most similar to what PINNs can in principle do) and published thermochronologic data from the region. Overall, this manuscript demonstrates that PINNs are capable of being trained to solve relatively simple crustal heat transfer problems, and they have potential for tackling more complex thermokinematic scenarios with more statistical rigor than what is presented in this contribution.
This manuscript provides an unprecedented perspective on how deep-learning approaches might be applied in thermochronology, and this is important because I think the next generation of computational tools we use to interpret cooling ages could benefit from such advances. By applying PINNs to the crustal thermal field problem, even with simple scenarios that our current tools are very capable of solving, this manuscript represents an important step in introducing deep-learning approaches to those of us interested in interrogating the thermal history of rocks using thermochronology. As such, with some revision for clarity (most of my suggestions are primarily provided as comments in the attached PDF), I think this contribution is a good fit for this journal. I do note that my expertise lies in thermal history and thermokinematic modeling of thermochron data, and not in neural networks, so I cannot really rigorously review the details of the PINN methods, functions, optimization, and learning rates described here. Instead, I focus my comments on the packaging of the method and how to make it accessible to a thermochron audience who might be considering using such tools.
Several specific comments that summarize my line-by-line feedback in the PDF:
1. This manuscript would be stronger if it clearly articulated why deep-learning approaches will (eventually?) be an improvement over current numerical methods. On the face of them, PINNs seem like huge black boxes — with “multiple hidden layers with trainable parameters and nonlinear activation functions” (lines 96-7). Make a more clear case for why pursuing deep-learning approaches like PINNs is worthwhile. What will we get? Why should I spend time learning about this? It seems clear from this work that these methods are still in the early stages of development, and have not yet superseded currently available methods. Or, if the authors think the PINN approach presented here is already an improvement over current methods in some way (computational time? accuracy?), they do not make a clear case for this. Either way, a more explicit argument for the utility of these methods would improve the impact of this paper.
I will also add that the synthetic 1D scenario and model design is set up such that the inversion is a pretty simple problem to converge on the “true” answer for, which limits the impact of the example a bit. A suite of 6 perfect thermochronologic ages that span a 400˚C temperature range produced by a simple two-stage monotonic cooling history in 1D is a very easy synthetic dataset to fit because it is distinctive — there is a narrow range of tT histories that can produce such a data set. This is especially true if, as is the case here, the inverse model design is narrow and dialed to only explore very similar/relevant histories (1D, monotonic, only exploring for one change in exhumation rate, only exploring 50 Ma of time, etc). Essentially, it’s not a hard problem to solve.
The synthetic 1D scenario presented here is an important starting demonstration, but many of the tools introduced in the beginning of this paper would perform similarly—no PINN black box required. I think a more interesting and compelling demonstration of the potential of this tool would be with a 1D (single-sample) synthetic dataset that is far less unique and more similar to a real dataset (fewer data) paired with a model design that explores whether these data could have been produced by a more complex or less constrained thermal history (i.e., starting earlier, non monotonic, multiple changes in exhumation rate possible). How well does this approach retrieve the “true” answer in this case? This is the realm of real thermochronologic datasets, and gets to the heart of the fundamental non-uniqueness of cooling ages, which is at the core of why thermochronologic studies rely on numerical tools for data interpretation.
The manuscript as it is already represents a complete contribution in my view, but I wanted to point out this additional opportunity.
2. Be more explicit about what is meant by “uplift” through out the paper (i.e., England and Molnar, 1990). The authors know that rock uplift only results in rock cooling if it is accompanied by exhumation, but that is not clear in the text as written. In the simple 1D scenarios, rock uplift = exhumation (both in rate and magnitude) because the surface elevation is not changing. But even then, using these terms interchangeably is not useful because there are many 1D scenarios where that would not be true. In the 3D example, the imposed rock uplift rate changes independently of the topographic evolution, so it is effectively decoupled from exhumation in space and time. Therefore, it is even more critical to be clear.
3. The model design and results are difficult to follow in places because it is hard to keep track of “model time” vs. “geologic time”, which are always opposites in such models. I suggest the authors can avoid some of this confusion by using “Myr” (millions of years) when referring to model time and timescales and “Ma” (millions of years ago) when referring to geological time and thermochronologic ages, in both the main text and figures.
4. The Dabie Shan example would benefit from some additional text describing the previous work (data, what is known about the orogenic history and timing) in more detail, so the model design decisions made have more geologic context.
5. The paper would benefit from a short conclusions section that summarizes the main findings and looks ahead more broadly to future applications of deep-learning approaches in thermochronology.
Ruohong Jiao et al.
Model code and software
PINNs for thermochronology Ruohong Jiao https://github.com/jiaor/PINNs_Chron/
Ruohong Jiao et al.
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