Thermodynamically Consistent Machine Learning for Fracture Prediction in Heterogeneous Materials

arXiv:2610.20614 · physics.comp-ph, cond-mat.mtrl-sci, cs.CE, cs.LG · Submitted 2026-07-12 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Thermodynamically Consistent Machine Learning for Fracture Prediction in Heterogeneous Materials".

Jane: The gist: The proposed tLaSDI-pGFINNs framework introduces a thermodynamics-informed,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we're looking at this paper today, "Thermodynamically Consistent Machine Learning for Fracture Prediction in Heterogeneous Materials." It tackles the problem of predicting how cracks grow in complex materials, which is super time-consuming with traditional physics-based simulations.

Jane: Right, so the basic idea seems to be using machine learning to make these predictions faster surrogates for when we can't run a full simulation. But what's new here is that they are building physics directly into the way the neural network learns.

Lu: Exactly, this paper is proposing something called tLaSDI-pGFINNs, which is a thermodynamics-informed approach using parametric GENERIC Formalism-Informed Neural Networks to learn the underlying state variables in a lower dimensional latent space.

Meng: That sounds fancy. So, instead of just learning patterns from data, they are forcing the model to follow laws like total energy conservation and entropy generation while it learns how the fracture evolves.

Lalam: Yeah, that constraint is pretty important because it means the predictions should be physically consistent even when we're testing on things we haven't seen before.

Tom: The paper focuses on using pGFINNs to represent parameter-dependent dynamics while making sure free energy conservation and entropy production hold across the whole parameter space. They construct operator matrices like the Poisson matrix L and the Friction matrix M to ensure specific properties for those latent states > Page two.

Jane: That sounds mathematically intensive. So, what's that geometry part coming in? I heard they also have a geometry-conditioned autoencoder involved in this tLaSDI-pGFINNs framework.

Lu: Yes, the paper develops a Geometry-Conditioned Autoencoder to learn the geometry configuration for effective learning and then incorporates that information into the decoder during reconstruction. They use something called Feature-Wise Linear Modulation, or FiLM, to inject geometry information into the decoder architecture > Page three.

Meng: So, it’s not just learning the material properties; they are also conditioning the prediction on how the material is physically shaped at that moment. That should help when you're dealing with really complex fracture paths.

Tom: The framework sets up a non-intrusive reduced-order model, which combines an autoencoder for dimensionality reduction with GFINN for latent dynamics prediction. They use a loss function that has four parts: integration loss, reconstruction loss, Jacobian loss, and modeling loss > Page three.

Title and authors: Jane: The integration loss sounds like it’s enforcing a consistency condition between the pGFINN’s integration and the encoder’s dynamics. That seems crucial for making sure the evolution over time is smooth and physically sound.

Lu: It's about ensuring that mu e(u n+one) about mu e(u n) + integral t n+one t n integral pGFINN(z(t), mu, t) dt > Page three.

Meng: From a deployment standpoint, if this works, it means we could potentially get these fracture predictions much faster than running the full physics simulations that usually take a lot of time.

Tom: And they showed some results on three different types of problems: single-cavity, single-inclusion, and multiple-inclusion problems. They compared their tLaSDI-pGFINNs against a purely data-driven Autoencoder Long Short-Term Memory model > Page three.

Jane: What did they find when they compared it to the standard AE-LSTM? Did this physics approach actually beat the purely data-driven one in terms of accuracy on unseen samples?

Tom: They did. In a single-cavity problem, tLaSDI-pGFINNs showed better generalization capability on unseen samples, with seventy percent of the samples predicted more accurately by the tLaSDI-pGFINNs compared to the AE-LSTM > Page three.

Meng: So it’s not just better accuracy on a test set; it’s better at handling new situations, which is what we actually care about when we deploy things in real materials.

Lu: And they confirmed that for the unseen testing samples, the energy and entropy histories over all time steps demonstrated conservation of energy and non-decreasing entropy production, confirming those thermodynamic constraints are satisfied > Page three.

Jane: That’s a big deal because it shows the model is not just fitting noise; it’s learning behavior that respects fundamental physical laws even when generalizing to new inputs.

Tom: When we compare the Mean Squared Error for the single-cavity problem, tLaSDI-pGFINNs achieved an MSE of one point eight seven seven x ten-three compared to the AE-LSTM's MSE of three point two three x ten-three > Page three.

Meng: That reduction in error is significant when you think about how much time we save by using this instead of a full-order simulation, even though it takes about five seconds per test case for prediction > Page three.

Jane: So, what does this mean for someone who just listens to the show? It means that when we talk about predicting how a crack will grow in something like concrete or metal, this approach is promising because it doesn't just learn from past examples; it learns *why* things happen based on energy and geometry.

Title and authors: Lu: The paper suggests that incorporating these thermodynamic constraints into the learning framework leads to more accurate, stable, and physically consistent predictions compared to models without embedded physical principles > Page three.

Tom: It really boils down to this: by training the pGFINN to learn the state variables of damage at every time step while inferring those energy and entropy conditions, it performs better than an LSTM that just uses nonlinear gating mechanisms without any physics guidance > Page three.

Jane: So, in a nutshell, the tLaSDI-pGFINNs framework is an approach that combines geometric conditioning with thermodynamic consistency to build a geometry-aware latent space dynamics identification method for accelerating fracture modeling in heterogeneous materials.

Meng: It’s solid work because they addressed the stability and physical consistency issues that plague many data-driven models when applied to these kinds of complex, path-dependent materials > Page three.

Lu: The way they integrate the Geometry-Conditioned Autoencoder with the pGFINN for thermodynamically consistent fracture evolution in the latent space is a key innovation here > Page three.

Lalam: I think it's a significant step because it moves beyond just pattern recognition into modeling the underlying physical processes of energy and entropy generation during damage propagation > Page three.

Tom: So, to wrap up, this paper demonstrates that incorporating thermodynamic constraints leads to more accurate and stable predictions compared to models without embedded physical principles > Page three.

Jane: It’s a solid piece of work for anyone trying to build reliable AI surrogates for engineering problems in materials science because it shows that physics guidance actually helps the machine learn better.

Lu: The way they use the pGFINN architecture to represent parametric latent dynamics while preserving free energy conservation and ensuring entropy production across the entire parameter space is a clever way to handle those constraints > Page two.

Meng: From a practical standpoint, if we can make these models robust enough for real-world applications, it means we could drastically cut down on the computational cost of testing new material designs > Page three.

Lalam: It's not just about speed; it’s about getting predictions that are reliable because they adhere to the fundamental rules of thermodynamics during the prediction process > Page three.

The paper's summary: Tom: So, to recap, this paper is about building an AI system that predicts how cracks form in complex materials by forcing the math inside the neural network to follow rules of physics like energy conservation and entropy generation.

Jane: Right, so instead of just looking at patterns in data points and guessing the next one, they are training the AI to understand the fundamental thermodynamic laws governing how damage happens during fracture.

Lu: The core idea is this tLaSDI-pGFINNs framework where they use these physics constraints through parametric GENERIC Formalism-Informed Neural Networks, or pGFINNs, to learn a low-dimensional latent space representation of the material's state variables.

Tom: So they are using those specific operator matrices like the Poisson matrix L and the Friction matrix M to make sure that when you look at that latent space, you're actually respecting free energy conservation and entropy production across all possible parameters.

Jane: That means if you feed it a new material setup, the predicted fracture path should still obey those laws, which is a big deal because most purely data-driven models just memorize the training examples and fail when things get slightly different.

Meng: From my side, I’m worried about what happens when we try to scale this up to something like a whole manufacturing process, but if the physics constraints hold true on unseen samples, it means the model is much more likely to be reliable in production environments.

Lu: And they didn't just stop there; they added a Geometry-Conditioned Autoencoder. This part helps them learn how the actual shape of the material influences where and how a crack goes, by injecting geometry information into the decoder using something called Feature-Wise Linear Modulation, or FiLM.

Tom: That means it’s not just learning the material's inherent properties; it’s learning how those properties change based on the physical configuration of the structure at that exact moment.

Jane: So if you were driving a car and you wanted to predict how a crack might propagate in a bridge, this model is trying to learn both what the bridge is made of and exactly where it's built in space simultaneously.

Meng: That conditioning sounds helpful for specific failure modes, but what about the computational cost? You mentioned earlier that inference takes about five seconds per test case. That’s still slow compared to a quick simulation run, even with physics embedded.

Tom: It is slow, sure, but they did show a speedup factor of roughly four hundred thirty-two times compared to running the full-order simulation for the single-cavity problem.

Jane: So it’s not instantaneous anymore, but it’s still way faster than waiting hours for a traditional model to finish calculating every little stress point.

Lu: The results on their testing—single cavities, inclusions, multiple inclusions—show that tLaSDI-pGFINNs actually beats the standard AE-LSTM in terms of accuracy on unseen samples, with seventy percent of those predictions being more accurate.

Tom: That’s a solid number. And they didn't just say it was better; they showed that the energy and entropy histories over all time steps confirm that these thermodynamic constraints are actually satisfied, even for samples they hadn't seen before.

Jane: That validation is pretty strong because it means the model isn't just cheating; it’s genuinely learning the underlying physical behavior of energy dissipation during crack growth.

Lu: The implication here is that we can move away from models that are just statistical fits and toward systems where the predictions are guaranteed to be physically plausible, which is a huge step for materials science AI.

Meng: If this approach works reliably across different material types, it could drastically cut down on the time and cost associated with testing new material designs before they even get to the physical lab.

Tom: So essentially, we’re taking complex physics—energy conservation, entropy generation—and weaving it directly into a deep learning architecture to create faster and more reliable fracture prediction tools.

The paper's improvements: Tom: So, we're looking at how they suggest improving this tLaSDI-pGFINNs framework for fracture prediction, moving beyond just getting a model that works okay on a test set.

Jane: What kind of improvements are they talking about? Are they tweaking the math or changing how the data is fed in?

Lu: They focus on refining the way the latent space dynamics identification works, specifically by making sure the training process is more robust to noise in heterogeneous materials.

Tom: The paper suggests tightening up those loss functions, especially that integration loss term, to make sure we get a better fit for how energy flows over time during fracture evolution.

Jane: So they are trying to enforce even stricter thermodynamic rules during the learning phase itself so the final prediction is cleaner and less sensitive to minor input variations.

Meng: From a deployment standpoint, that sounds good because it means if we feed it slightly noisy sensor data from a real-world material test, the model shouldn't just break or give us garbage results.

Lu: Exactly; they are looking at how to handle uncertainty better by adding terms that quantify the epistemic uncertainty in their predictions through something called Credal Geometry.

Tom: That means when the AI predicts a fracture pattern, it can tell you not just what it thinks is happening, but also how sure it is about that prediction based on what’s missing in its training data.

Jane: That adds a layer of honesty to the prediction, which is important because we don't want an AI confidently predicting something that might be totally wrong if it hasn't seen that exact scenario before.

Meng: But what about the geometry part? They mentioned using FiLM modulation, and they suggest exploring more ways to condition that decoder to get even better geometry-sensitive fracture features.

Lu: They are looking at ways to make that conditioning more adaptive so the model can focus its attention on the most relevant geometric details during reconstruction, rather than just applying a fixed scaling factor.

Tom: It’s about making sure the geometry information is used intelligently, not just tacked onto the architecture; they want that FiLM layer to be truly guiding it toward those critical fracture features.

Jane: So if we think of it like driving, the first part is having a good map and knowing where you are—that’s the geometry conditioning—and now they want a better steering system for when you hit a tricky road.

Meng: That makes sense for real-world deployment; if the model can better handle geometric variations, it means we don't have to spend so much time manually labeling or correcting complex three dee structural data.

Lu: Their goal is to create a system that not only predicts the pattern accurately but also provides reliable uncertainty estimates based on how much information the material structure provides at that moment.

Tom: So it’s not just about getting a lower MSE number; it’s about building an AI surrogate that can provide physically consistent and trustworthy predictions even when those predictions are slightly outside its training distribution.

Conclusion: Tom: So to wrap up, this paper "Thermodynamically Consistent Machine Learning for Fracture Prediction in Heterogeneous Materials" shows that adding thermodynamic constraints—like energy conservation and entropy generation—into the learning process leads to predictions that are much more stable and physically consistent than standard models.

Jane: It really proves that forcing the AI to respect those fundamental laws of physics makes it a much better tool for engineering problems, especially when you're dealing with materials that aren't perfectly uniform.

Lu: The implication is we move toward AI surrogates that are not just statistical mimics but systems that respect the underlying physical mechanisms of damage propagation in heterogeneous media.

Meng: For production, this means models built on this framework should be much less likely to fail catastrophically when encountering new material conditions because they have built-in physical guardrails.

Lalam: I'm confident in this answer; the paper shows that physics guidance actually improves generalization on unseen samples and that the conservation laws are maintained throughout the entire prediction process.

Tom: And we saw concrete numbers, like achieving a mean squared error of one point eight seven seven times ten to minus three for the single-cavity problem, which is significantly lower than the baseline AE-LSTM model's error.

Jane: So when you listen to this, think about how much time you save on running those massive physical simulations just to get a rough idea of how a crack might grow.

Lu: The real impact is on materials science; it opens the door for creating more reliable and faster predictive tools for complex structural integrity problems.

Meng: It's not instant, though; we still have that five seconds per test case for inference, but that speedup over a full-order simulation is substantial enough to justify the extra training time.

Tom: And they did mention that the Geometry-Conditioned Autoencoder helps condition the reconstruction process with geometry information using FiLM modulation, which sharpens those fracture pattern predictions.

Jane: That means when we use this system, it doesn't just predict a crack; it predicts *where* and *how* based on the actual shape of what we’re testing.

Lu: The future work they hint at involves pushing the limits of how well these constraints hold up as the complexity of the material structure increases, which is where we need to look next.

Meng: I'm still focused on deployment, so I'd be interested to see if they can make this inference time drop even further without losing those physical guarantees.

Lalam: It’s a solid direction; the path forward is clearly about scaling this thermodynamic consistency into even more complex, real-world scenarios.

University of California San Diego Department of Mechanical and Aerospace Engineering · Computational and Multiscale Mechanics Group Synopsys Inc. · National Cheng Kung University Department of Civil Engineering · National Cheng Kung University Department of Engineering Science · University of California San Diego Department of Structural Engineering

physics.comp-ph, cond-mat.mtrl-sci, cs.CE, cs.LG

Submitted: 2026-07-12

Updated: 2026-07-12

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 73/100

The gist: The gist: The proposed tLaSDI-pGFINNs framework introduces a thermodynamics-informed, geometry-conditioned latent space dynamics identification approach to predict fracture patterns and evolution in

Key concepts

tLaSDI: Thermodynamically Consistent Latent Dynamics Identification is the core method that learns the thermodynamic state variables within a compressed, low-dimensional latent space. It uses pGFINNs to ensure that these learned states adhere strictly to physical laws such as total energy conservation and entropy generation throughout the simulation.
pGFINNs: Parametric GENERIC Formalism-Informed Neural Networks are specialized neural networks designed to enforce thermodynamic principles directly into their architecture. By constructing specific operator matrices, they ensure that the learned dynamics preserve key physical constraints, making them physically consistent predictors.
Geometry-Conditioned Autoencoder (GC-AE): This component learns the geometric configuration of the material structure. It uses Feature-Wise Linear Modulation (FiLM) to inject geometry information into the decoder, allowing it to guide the reconstruction process from latent space back to physical space by amplifying features sensitive to specific material shapes.

Summary

The gist: The proposed tLaSDI-pGFINNs framework introduces a thermodynamics-informed, geometry-conditioned latent space dynamics identification approach to predict fracture patterns and evolution in heterogeneous materials by learning thermodynamic state variables in a lower-dimensional latent space using parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) that enforce key thermodynamic principles like total energy conservation and entropy generation.

Thermodynamic Consistency via pGFINNs

The core of the model is the Thermodynamically Consistent Latent Dynamics Identification (tLaSDI) for parametric systems, which uses parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) to predict fracture patterns and evolution in heterogeneous materials. This approach embeds physics into the neural network architecture using a hard-constraint method that inherently complies with the underlying thermodynamic principles, such as total energy conservation and entropy generation. The pGFINNs are parameterized by M to effectively represent parameter-dependent dynamics while preserving free energy conservation and ensuring entropy production across the entire parameter space. This is achieved by constructing operator matrices, such as the Poisson matrix L and the Friction matrix M, which are designed to ensure ∇zS ∈ ker(L) and ∇zE ∈ ker(M), respectively.

Geometry-Conditioned Autoencoder for Reconstruction

The tLaSDI-pGFINNs framework is further enhanced by a Geometry-Conditioned Autoencoder in learning the geometry configuration for effective geometry-guided learning. The Geometry Conditioned Autoencoder (GC-AE) is proposed to guide the reconstruction process from the latent space to the physical space by incorporating geometry information during decoding process. Specifically, geometry information is injected into the decoder architecture using the Feature-Wise Linear Modulation (FiLM) method, which was originally implemented to enhance the visual reasoning task that is challenging for standard deep learning architectures. This FiLM modulation adaptively scales and shifts feature activations through parameters gammai,c and betai,c, enabling the decoder to amplify geometry-sensitive fracture features during training.

tLaSDI-pGFINN Formulation and Loss Function

The tLaSDI-pGFINN framework models high-dimensional dynamical systems using a non-intrusive ROM that combines an autoencoder for nonlinear dimensionality reduction with GFINN for latent dynamics prediction. The inference mode involves encoding the high-dimensional initial state u0 into a low-dimensional latent representation z0 using the convolutional encoder phie, and then using the pGFINN to predict thermodynamically consistent latent states at subsequent time steps. The loss function of tLaSDI-pGFINNs consists of four terms: integration loss (int), reconstruction loss (rec), Jacobian loss (jac), and modeling loss (mod). The integration loss enforces the approximate equality phie(un+1) ≈ phie(un) + ∫tn+1 to tn ∫ħpGFINN(z(t), mu, t) dt, providing a consistency condition between pGFINN integration and the encoder dynamics.

Performance and Validation

The proposed framework was applied to three numerical examples involving varying levels of complexity: (1) a single-cavity problem with varying loading conditions, (2) a single-inclusion problem with varying inclusion locations and diameters, and (3) a multiple-inclusion problem with varying crack initiation locations. The results demonstrate that tLaSDI-pGFINNs can more accurately capture both fracture patterns and the subsequent fracture evolution than the purely data-driven Autoencoder Long Short-Term Memory (AE-LSTM) model. For instance, in a single-cavity problem, tLaSDI-pGFINNs showed better generalization capability on unseen samples, with 70% of the samples predicted more accurately by the tLaSDI-pGFINNs compared to the AE-LSTM. Furthermore, energy and entropy histories over all time steps of the tLaSDI-pGFINN model demonstrate conservation of energy and non-decreasing entropy production, confirming that the thermodynamic constraints are satisfied even for unseen testing samples.

Comparison with Baseline Model

When comparing the models in a single-cavity problem, Table 1 illustrates that tLaSDI-pGFINNs achieved a lower Mean Squared Error (MSE) of 1.877 × 10−3 compared to the AE-LSTM's MSE of 3.23 × 10−3. This superior performance is attributed to the fact that pGFINN is trained to learn the state variables of the damage at every time step during training by inferring the energy and entropy conditions, whereas the LSTM relies on nonlinear gating mechanisms without physics guidance. The tLaSDI-pGFINN model also requires approximately 5 seconds per test case for prediction, achieving a speedup of approximately 432 times compared to the full-order simulation.

Conclusion

The study concludes that incorporating thermodynamic constraints into the learning framework leads to more accurate, stable, and physically consistent predictions compared to models without embedded physical principles. The tLaSDI-pGFINN model successfully predicts crack patterns that closely match ground truth solutions and exhibits better generalization capability on unseen samples due to the enforcement of underlying physical behavior governing fracture evolution. This behavior is further supported by the energy and entropy histories obtained from unseen testing samples, where both conservation and dissipation laws remain satisfied throughout the prediction process.

How it works

The model learns the thermodynamic state variable in a lower-dimensional latent space using pGFINNs that enforce key thermodynamic principles, such as total energy conservation and entropy generation.
A Geometry-Conditioned Autoencoder is used to learn the geometry configuration for effective geometry-guided learning.
The tLaSDI-pGFINNs framework integrates these components to develop a thermodynamics-informed, geometry-conditioned latent space dynamics identification framework for accelerated, geometry-aware fracture modeling in heterogeneous materials.
The training utilizes a comprehensive loss function consisting of integration loss, reconstruction loss, Jacobian loss, and modeling loss to ensure joint training of the autoencoder and the latent dynamics model.
In inference mode, the high-dimensional initial state is encoded into a low-dimensional latent representation using the convolutional encoder phie, and then the pGFINN predicts thermodynamically consistent latent states at subsequent time steps.
The Geometry Conditioned Autoencoder uses FiLM layers to modulate feature activations, guiding the decoder to amplify geometry-sensitive fracture features during training.

Key Findings

((3) The model learns the thermodynamic state variable in a lower-dimensional latent space using parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) that enforce key thermodynamic principles, such as total energy conservation and entropy generation.

((5) The model learns the thermodynamic state variable in a lower-dimensional latent space using parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) that enforce key thermodynamic principles, such as total energy conservation and entropy generation.

**((7) The proposed framework integrates geometry-conditioned convolutional autoencoders for nonlinear dimensionality reduction with a pGFINN for thermodynamically consistent fracture evolution in the latent space.

Improvements for AI systems

A thermodynamic-consistent latent dynamics identification framework (tLaSDI-pGFINN) can be used to predict fracture patterns and evolution in heterogeneous materials by learning "the thermodynamic state variable in a lower-dimensional latent space using parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) that enforce key thermodynamic principles, such as total energy conservation and entropy generation."
The integration of a Geometry-Conditioned Autoencoder (GC-AE) enhances the model's ability to capture geometry-sensitive fracture features by inject[ing] geometry information into the decoder architecture using the Feature-Wise Linear Modulation (FiLM) method, leading to sharper and more accurate crack pattern predictions than those generated by AE-LSTM.
The resulting improved AI system can perform high-accuracy, physics-informed surrogate modeling for fracture prediction in complex materials with heterogeneous properties, specifically demonstrating superior generalization capability on unseen testing samples compared to purely data-driven models.