Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition
- Life and Environemental Science Department, University of California, Merced, Merced, CA, USA
- Life and Environemental Science Department, University of California, Merced, Merced, CA, USA
Abstract. Modeling water flow in unsaturated soils is vital for describing various hydrological and ecological phenomena. Soil water dynamics is described by well-established physical laws (Richardson-Richards equation (RRE)). Solving the RRE is difficult due to the inherent non-linearity of the processes, and various numerical methods have been proposed to solve the issue. However, applying the methods to practical situations is very challenging because they require well-defined initial and boundary conditions. Recent advances in machine learning and the growing availability of soil moisture data provide new opportunities for addressing the lingering challenges. Specifically, physics-informed machine learning allows taking advantage of both the known physics and data-driven modeling. Here, we present a physics-informed neural networks (PINNs) method that approximates the solution to the RRE using neural networks while concurrently matching available soil moisture data. Although the ability of PINNs to solve partial differential equations, including the RRE, has been demonstrated previously, its potential applications and limitations are not fully known. This study conducted a comprehensive analysis of PINNs and carefully tested the accuracy of the solutions by comparing them with analytical solutions and accepted traditional numerical solutions. We demonstrated that the solutions by PINNs with adaptive activation functions are comparable with those by traditional methods. Furthermore, while a single neural network (NN) is adequate to represent a homogeneous soil, we showed that soil moisture dynamics in layered soils with discontinuous hydraulic conductivities are correctly simulated by PINNs with domain decomposition (using separate NNs for each unique layer). A key advantage of PINNs is the absence of the strict requirement for precisely prescribed initial and boundary conditions. In addition, unlike traditional numerical methods, PINNs provide an inverse solution without repeatedly solving the forward problem. We demonstrated the application of these advantages by successfully simulating infiltration and redistribution constrained by sparse soil moisture measurements. As a free by-product, we gain knowledge of the water flux over the entire flow domain, including the unspecified upper and bottom boundary conditions. Nevertheless, there remain challenges that require further development. Chiefly, PINNs are sensitive to the initialization of NNs and are significantly slower than traditional numerical methods.
- Preprint
(4596 KB) -
Supplement
(1658 KB) - BibTeX
- EndNote
Toshiyuki Bandai and Teamrat A. Ghezzehei
Status: final response (author comments only)
-
RC1: 'Comment on hess-2022-73', Silvio Gumiere, 24 Mar 2022
Comments on: Forward and inverse modelling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition.
Summary
This paper presents the results from a comprehensive study using PINNs as a forward and inverse numerical solution for the Richardson-Richards equation. They tested new approaches for applying the PINN method, including a layer-wise locally adaptative function intended to work with layered heterogeneous soil profiles. In addition, the authors compared their approach to well-known numerical solutions for the Richardson-Richards equation, namely Finite Difference and Finite Element Methods (FDM and FEM). The PINNs approach was also validated with soil moisture measurements performed in a soil column in controlled conditions.
The paper appears to be relatively novel, being the first application of PINNs to the Richardson-Richards equation (to my knowledge). The literature proposed and the figures presented are of high quality. I enjoyed reading the paper. The results are encouraging on the applications of PINNs to model hydrodynamics in porous media, even if it takes much more time when compared with the classical approaches. We don't need to impose well-known boundaries and initial conditions, which is attractive once they are difficult to obtain in field applications. The domain decomposition for the layered soils is also very promising. Even classical approaches such as FDM and FEM struggle with heterogeneous soil profiles. So, I think the PINNs with the domain decomposition did quite well in modelling the soil water dynamics in the soil column.
Specific comments/questions (that should be addressed and commented before publication):
- It would be interesting to test the inverse solution with soil matric potential measurements (data is available if needed).
- What is your opinion on going to 2 and 3D modelling? Could the domain decomposition proposed in the paper be applied to speed up 2 and 3D solutions? I think that would be the actual gain in this methodology. FEM applications for the fully 3D solution of Richardson-Richard's equation are still slow and have many complications with mesh, especially for large domains. This also applies to the boundary and initial conditions imposition.
- What about non-Darcian conditions, macropore flow, very high clay content soils. Do you think the method could be applied?
- What about root-water-uptake? How can this be included in your approach? There exist some analytical solutions for these problems (Yuan and Lu, 2005[1])
- Do you think one day the PINNs could take over the classical approaches? What is limiting it?
- What about practical applications? Irrigation management or contaminant transport in the vadose zone.
Overall, the paper is well written. The sections are balanced, and the flow is good, making the paper enjoyable to read.
[1] Yuan, Fasong, and Zhiming Lu. "Analytical solutions for vertical flow in unsaturated, rooted soils with variable surface fluxes." Vadose Zone Journal 4.4 (2005): 1210-1218.
-
AC1: 'Reply on RC1', Toshiyuki Bandai, 30 Mar 2022
We appreciate the constructive comments and feedback by Silvio Gumiere. Our replies to the comments and questions are in the supplemental document attached.
- AC3: 'Reply on AC1', Toshiyuki Bandai, 07 Apr 2022
-
RC2: 'Comment on hess-2022-73', Anonymous Referee #2, 04 Apr 2022
Review of “Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition” by Bandai and Ghezzehei.
In this manuscript the authors tested a physics-informed neural networks (PINNs) method to solve the Richardson-Richards equation for simulating unsaturated soil water dynamics. The authors also investigated the capability of the method for obtaining inverse solutions. As coupling data-driven and physics-based approaches have received much attention these days, the topic fits well with the scope of HESS. The authors have done a great job on demonstrating how PINNs performed when simulating unsaturated water flow in soils and showing applicability and limits of the method. Although the paper was well organized and written, I believe the paper has a room for some improvement. I have some comments that should be addressed prior to accepting this paper for publication. For my curiosity, I am wondering if this approach can be applied to simulate preferential type flow in soils. Is it going to be straightforward? Does it require some modifications in the model? If it can be applied to such phenomena, it would be a great breakthrough in the field of soil physics and hydrology.
General comments:
In Fig. 5, the evolution of PINNs solution is plotted. At the initialization, some of the solutions are beyond the limit of the water content as the water content values are greater than the saturated water content. Would it be possible to put some constraints to the solutions in PINNs? If so, would that improve training and overall performance? Any discussions on this matter will helpful for those who are interested in using this method. A similar question goes to the inverse solutions. I am wondering if any constraints can be applied to the target parameters that are inversely estimated. There is always a need to put some constrains to the parameters being estimated.
In the demonstration of getting inverse solution with PINNs, the authors used a 2-layered soil system. Why? If the boundary fluxes are being estimated, wouldn’t be better to start with a homogenous case? Was there a specific reason that the layered soil system was used in this demonstration?
Specific comments:
L189: It sounds a bit strange to say that soil dynamics is “controlled by the volumetric water content at the bottom.”
L193 (Eq.15): A little bit more explanations will be helpful to understand this transformation. I have no idea why the beta value gives better initial guess.
L311: If the logarithmic transformation of water potential is used, the approach is limited to “unsaturated” systems. But there are many cases you will have both positive and negative potential values. How do you deal with that?
Figure 3(b): There are some systematic differences between FDM and PINNs. Why? Are these because of the choice of spatial and temporal discretization in FDM?
Figure 10: Looks like something is wrong with the texts at the top of the figures.
Figure 16: For all three cases, the PINN solutions show that the inversely estimated initial surface flux is much smaller than the true flux. Are there any specific reasons for this?
-
AC2: 'Reply on RC2', Toshiyuki Bandai, 07 Apr 2022
We appreciate your constructive comments and feedback. Our replies to the comments and questions are in the supplemental document attached.
- AC4: 'Reply on AC2', Toshiyuki Bandai, 07 Apr 2022
-
AC2: 'Reply on RC2', Toshiyuki Bandai, 07 Apr 2022
-
RC3: 'Comment on hess-2022-73', Anonymous Referee #3, 22 Apr 2022
The paper is very interesting and introduces a physics-informed neural networks (PINNs) method in a Richards’ equation context, aimed at approximating the solution to the RRE using neural networks while concurrently matching available soil moisture data. In particular, in this paper authors consider domain decomposition for handling infiltration into layered soils.
The topic is definitely up to date, the paper is well written, and provides all the details for implementing and understanding this approach. Nevertheless, I think authors should address some comments and issues before it can be accepted.
- This PINN approach appears really fascinating because it allows to integrate physics-based models (such as RRE) with machine learning features. Authors ascribe the uncertainties in Richards’equation to the choice of boundary conditions, which is surely right. Nevertheless, I think they do not consider the (even more) cumbersome uncertainties arising from the choice of model parameters, which are the result of some non-linear fitting in laboratory experiments (I am referring to the parameters in the WRC and HCF). This point is a main concern for me: as a matter of facts, unsaturated flow dynamics strongly relies on functions parameters, rather than on ICs and BCs, which are generally easier to assess. On the other hand, I see authors already published a paper on this topic: I think it would be valuable to stress the differences between the two papers
- Lines 197-198: few more words for sketching how the partial derivatives are computed would be valuable
- Figure 1: I think there is a typo in the box “Physics and Data Constraints”, since the partial derivative at the left-hand side should be accomplished with respect to time.
- I understand that the residual is computed between the synthetic data and the computed (by the PINNs) ones; in this framework, what is the rationale of comparing the PINNs output with any Richards solver (as Hydrus)?
- As far as I understand, the power of this approach is to combine physics-based models with data driven ones; according to my knowledge, this is also the spirit of Data Assimilation (DA) methods, which incorporate measurements into a physics based model, albeit in a very different framework; these methods have also been treated in Richards’ equation context (see for instance Berardi et al CPC https://doi.org/10.1016/j.cpc.2016.07.025, Medina et al HESS https://doi.org/10.5194/hess-18-2521-2014, Liu et al JoH https://doi.org/10.1016/j.jhydrol.2020.125210 ); what is authors’ opinion about this? What would be the pros and cons of PINNs approach with respect to DA one? Also DA methods allow to assimilate boundary conditions, as in this case, and hydraulic parameters, as well as states. As a matter of fact, with respect to DA methods, this PINNs approach seems to me more on the theoretical side (which is definitely fine, of course) rather than application oriented.
- Authors mention the possibility to drop loss terms for IC or BC at line 225. However, they have not presented any experiment for this scenario. Could you please comment on this ill-posed configuration? How would it perform with respect to classical solver?
- Figure 11 and 3. Please replace “Fintie” with “Finite”.
- Authors make use of synthetic data: I had hard times to find where the reference to used data is described. Of course the use of synthetic data is fine, but they should highlight it at the beginning of the paper. Moreover, could you please explain how your method of synthetic data generation could compare to real measurement data? In other words, how robust is your result with respect to outliers, sensor noise and other technical issues when it comes to real data?
- AC5: 'Reply on RC3', Toshiyuki Bandai, 21 May 2022
Toshiyuki Bandai and Teamrat A. Ghezzehei
Data sets
DD-PINNS-RRE Toshiyuki Bandai and Teamrat A. Ghezzehei https://doi.org/10.5281/zenodo.6030635
Model code and software
DD-PINNS-RRE Toshiyuki Bandai and Teamrat A. Ghezzehei https://doi.org/10.5281/zenodo.6030635
Toshiyuki Bandai and Teamrat A. Ghezzehei
Viewed
HTML | XML | Total | Supplement | BibTeX | EndNote | |
---|---|---|---|---|---|---|
550 | 153 | 25 | 728 | 53 | 7 | 7 |
- HTML: 550
- PDF: 153
- XML: 25
- Total: 728
- Supplement: 53
- BibTeX: 7
- EndNote: 7
Viewed (geographical distribution)
Country | # | Views | % |
---|
Total: | 0 |
HTML: | 0 |
PDF: | 0 |
XML: | 0 |
- 1