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Authors
Yubo Ye, Huafeng Liu, Xiajun Jiang, Maryam Toloubidokhti, Linwei Wang
Abstract
Physics-informed neural network (PINN) is a new paradigm for solving the forward and inverse problems of partial differential equations (PDEs). Its penetration into 3D bi-ventricular electrophysiology (EP) however has been slow, owing to its fundamental limitations to solve PDEs over large or complex solution domains with sharp transitions. In this paper, we propose a new PINN framework to overcome these challenges via three key innovations: 1) a weak-form PDE residual to bypass the challenges of high-order spatial derivatives over irregular spatial domains, 2) a spatial-temporally adaptive training strategy to mitigate the failure of PINN to propagate correct solutions and accelerate convergence, and 3) a sequential learning strategy to enable solutions over longer time domains. We experimentally demonstrated the effectiveness of the presented PINN framework to obtain the complete forward and inverse EP solutions over the 3D bi-ventricular geometry, which is otherwise not possible with vanilla PINN frameworks.
Link to paper
DOI: https://doi.org/10.1007/978-3-031-43990-2_16
SharedIt: https://rdcu.be/dnwLr
Link to the code repository
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Link to the dataset(s)
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Reviews
Review #3
- Please describe the contribution of the paper
The work focused on physics-informed neural networks (PINNs) designed to solve the forward and inverse 3D bi-ventricular electrophysiology problem. A two-variable Aliev-Panfilov diffusion-reaction model is considered. PINNs consist in training neural networks to satisfy physical laws described by PDEs and optimized via PDE residuals. It’s an interesting subject. A variational formulation is used to decrease the order of spatial derivatives. A spatiotemporally adaptive training strategy is then proposed to allow solutions over a large time period and to improve convergence. Contrary to the published literature, a 3D bi-ventricular domain is used in this work. Some capabilities of the introduced method are highlighted through experimental validation.
- Please list the main strengths of the paper; you should write about a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
- The numerical methodology is original using PINNs and working on realistic complex 3D geometry (or specific to the patient).
- The use of the weak formulation to decrease the order of differentiability.
- The use of the spatio-temporal adaptive training strategy.
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
The paper is overall well written. The methodology has enough novelties and is clearly presented. See answer to question 3. However, I find the numerical result part to be a bit compact. It would be interesting to evaluate the precision of the code from a numerical point of view (convergence of the mesh, convergence in time and orders of convergence, …). A comparative study of the computational cost is carried out. I also understand that authors have certain page limitations and can best expand on this part in a journal article. Overall I thought it was a nice job.
- Please rate the clarity and organization of this paper
Excellent
- Please comment on the reproducibility of the paper. Note, that authors have filled out a reproducibility checklist upon submission. Please be aware that authors are not required to meet all criteria on the checklist - for instance, providing code and data is a plus, but not a requirement for acceptance
I have a few minor comments.
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I recommend the authors to better describe the time-stepping strategy and to further justify this choice in relation to the existing adaptation strategies.
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I can’t find the temporal discretization strategy. For several schemes (eg BDF-2), the use of a variable time step will affect the coefficients multiplying the different approximations. The authors should clarify this point.
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Typo on page 5, paragraph 2 “is acehived”.
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- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review: https://conferences.miccai.org/2023/en/REVIEWER-GUIDELINES.html
Please see my answers to questions 3, 4 and 5.
- Rate the paper on a scale of 1-8, 8 being the strongest (8-5: accept; 4-1: reject). Spreading the score helps create a distribution for decision-making
6
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
NA
- Reviewer confidence
Confident but not absolutely certain
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
N/A
- [Post rebuttal] Please justify your decision
N/A
Review #2
- Please describe the contribution of the paper
The authors propose a method to perform forward and inverse PDE modeling. They base their method on physics-informed neural networks (PINN). They introduce a few startegies to overcome limitations of PINNs to solve PDEs over large or complex solution domains with sharp transitions. They test the method on cardiac electrophysiology coupled diffusion-reaction model system of equations.
- Please list the main strengths of the paper; you should write about a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
The authors introduce a few reasonable heuristics to facilitate convergence of PINNs to the ground truth PDE solution. Namely, they propose to add two loss weighting components:
1) temporal weights to dynamically focus on a correct PDE solution at time t after ensuring that solutions at previous (1,..,t-1) time steps are of sufficient accuracy (equally weighted simultaneous optimisation of all solution (1, …, t) is prone to failure); 2) spatial weights to focus optimisation in areas of sharper gradients. Finally, motivated by the same reason due to which the temporal weights were introduced - challenging nature of training PINNS for arbitrarily long time intervals - they propose a sequential training scheme. In this scheme, the whole time interval is divided into subintervals. A PINN is trained sequentially across this subintervals. A solution from previous time is used for training PINN and obtaining a solution at the next time step.
Sufficient ablation analysis is presented showcasing contirubtion of the introduced heuristics.
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
The method is not tested on real data. The method is compared against a PINN solution, and parametric inference is tested on synthetic data with known parameters. As Fig. 5 shows, while the MSE error for the inverse problem is low, the predicted solution simualtion profile can visually be quite different from the true solution. And this is only synthetic cases, in the case of real data there will be noise introduced in the training through the data residual loss and performace is expected to be even worser. As MICCAI is an applied, clinically oriented venue, a clear potential for a working in practice solution is expected.
- Please rate the clarity and organization of this paper
Very Good
- Please comment on the reproducibility of the paper. Note, that authors have filled out a reproducibility checklist upon submission. Please be aware that authors are not required to meet all criteria on the checklist - for instance, providing code and data is a plus, but not a requirement for acceptance
The method proposed in the paper looks reproducible.
- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review: https://conferences.miccai.org/2023/en/REVIEWER-GUIDELINES.html
Please comment on weaknesses above.
Your solution takes up to 1h of computing time, Tab.1. For reference, can you provide timings using the numerical solution?
Minor typos:
- in Eq.7 for Loss_I, I assume h(x_i) is u_theta (x_i, 0). If not, explain what is h()
- In Eq.7 for Loss_B, in the nominator u should go with subscript theta?
- in Eq. 15, I assume the second term is Res2,k and not Res1,k ?
- propgate
- acehived
- Rate the paper on a scale of 1-8, 8 being the strongest (8-5: accept; 4-1: reject). Spreading the score helps create a distribution for decision-making
5
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
Even though the propose heuristics are sensible, the method is tested on synthetic data and already on this data exhibits notable misalignment with the true solution. In the case of real data, there will be noise that would likely worsen the picture.
- Reviewer confidence
Confident but not absolutely certain
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
N/A
- [Post rebuttal] Please justify your decision
N/A
Review #1
- Please describe the contribution of the paper
The paper presents a model of 3D biventricular electrophysiology based on physics-informed neural networks. The authors describe an extension of the PINN approach to better represent complex spatiotemporal dynamics typical of cardiac electrophysiology. The key novelties introduced by the paper are three. The loss term representing the residual of the forward model partial differential equation is formulated based on a weak form of the equations, to overcome difficulties in approximating high order derivatives. The loss terms are corrected during training to promote the propagation of nontrivial solutions as well as solutions with sharp gradients, to expose the network to a larger number of such examples. The training process is subdivided in multiple temporal intervals to help the network produce solutions over longer time spans. The framework is validated by assessing the accuracy of the solution both for the forward and the inverse electrophysiology problem.
- Please list the main strengths of the paper; you should write about a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
The novel elements introduced by this paper to the standard PINN framework are clearly explained, well motivated and clever. The description of the method is reasonably detailed and multiple aspects of the performance are considered (accuracy and computational cost).
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
Some key information seems to be missing to ensure proper reproducibility of the results and proper assessment of the method. First of all, the method seem to require that the neural network is re-trained for each new geometry and parameter combination in equations 1-3. This should be clearly stated, because it implies that for each new data set to be studied, one forward solution has to be computed with a standard numerical method (to provide the ground truth); after which, the training procedure is executed, with additional computational time as listed in Table 1. In this scenario, the utility of the PINN network is limited to the application for the solution of the inverse problem. In any case, it would be useful to report for comparison the compute time of the numerical method used to produce the ground truth data.
The focus of the paper seems therefore to be on the technical challenges of designing an efficient PINN framework to create an electrophysiology model, more than on the practical applications of the framework itself. While I think this is an important topic, I also think the paper should devote some more space to discussing the practical implications of the availability of a more accurate PINN framework, in particular in the context of data assimilation. The example considered in the paper (retrieving parameter gamma from available TMP solutions) is simplistic and not properly develop into a use case with practical / clinical utility.
- Please rate the clarity and organization of this paper
Very Good
- Please comment on the reproducibility of the paper. Note, that authors have filled out a reproducibility checklist upon submission. Please be aware that authors are not required to meet all criteria on the checklist - for instance, providing code and data is a plus, but not a requirement for acceptance
The method is overall clearly explained. Some additional details would be needed for a practical implementation: in particular, what are the requirements for the numerical simulation (what kind of geometry representation / computational grid, temporal and spatial resolution).
- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review: https://conferences.miccai.org/2023/en/REVIEWER-GUIDELINES.html
Overall: the numerical results look qualitatively good, but it’s difficult to interpret them without more precise description. What is the unit of measurement of time in the different figures? Time goes from 0 to 2 in figure 3, from 10 to 60 in figure 4, from 0 to 20 in Table 1. Similarly for variable u, between 0 and 1 in figure 4 and between 0.5 and 1 (numerical solution) in figure 3.
Given the visual results in Fig. 5, arguably MSE may not be the best metric to use to assess accuracy for the parameter identification scenario. Max error could be more informative, since it looks like the method can incorrectly estimate relatively high gamma in regions in which the ground truth value is small. Along the same line of reasoning, it would be interesting to assess the error in the solution in areas with high gradients, such as the regions marked by high values of the spatial weights in Fig 4.
Section 3.2: propgate –> propagate acehived –> achieved
Fig 3. Where exactly is this solution measured, and what is the measurement unit of the time axis? The temporal dynamics is quite unexpected for a transmembrane potential.
What data precisely was used to train (looks like it’s trained on a single geometry and using a single set of parameters?)
- Rate the paper on a scale of 1-8, 8 being the strongest (8-5: accept; 4-1: reject). Spreading the score helps create a distribution for decision-making
5
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
The approach is interesting and the topic is important. I think this paper should be presented and discussed at the conference. I have some reservations about the description of the method and results as detailed in my comments to the authors. I also think that in its current form, the paper is too focused on the technical challenges to be overcome to make the method work. The practical utility of this approach for specific use cases of potential clinical relevance should be discussed.
- Reviewer confidence
Confident but not absolutely certain
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
N/A
- [Post rebuttal] Please justify your decision
N/A
Primary Meta-Review
- Please provide your assessment of this work, taking into account all reviews. Summarize the key strengths and weaknesses of the paper and justify your recommendation. In case you deviate from the reviewers’ recommendations, explain in detail the reasons why. In case of an invitation for rebuttal, clarify which points are important to address in the rebuttal.
This paper presents a physics-informed neural network (PINN) framework for spatiotemporal simulations and parameter inference for 3D biventricular electrophysiology. Instead of using the strong-form PDEs which can be difficult for PINNs, the weak-form PDEs with spatial discretization are used. Spatial-temporally adaptive strategies are also introduced to improve the training. As mentioned by the reviewers, this paper is well written, and the framework proposed is reasonable and innovative. On the other hand, experiments were only performed on simulation but not real data. Apart from the issues mentioned by the reviewers, the authors should also provide more details on how the numerical ground truths were generated (e.g., FDM, FEM?). The time required to generate the ground truths should also be shown, as implied in the introduction, a reason of using PINNs instead of traditional numerical methods is the computational advantage.
Author Feedback
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