Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning

Alfio Quarteroni; Francesco Regazzoni; Paola Gervasio

REVIEW 2 major objections 6 minor 2 cited by

Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Scientific Machine Learning—hybrid models that inject physics into neural networks and use data to enrich PDE solvers—is, this survey argues, a mature and effective strategy for complex problems governed by partial differential equations…

desk verdict A competent, well-referenced survey that will help newcomers, but it repeats the common high-dimensional PINN overstatement that its own complexity theorems contradict, and it presents the authors' prior cardiac results as established without benchmarks. read the letter →

arxiv 2501.18708 v2 pith:LVZQ25WO submitted 2025-01-30 math.NA cs.LGcs.NAphysics.comp-ph

Alfio Quarteroni , Paola Gervasio , Francesco Regazzoni This is my paper · ORCID

classification math.NAcs.LGcs.NAphysics.comp-ph MSC 68T0765M60

keywords ScientificMachineLearningphysics-informedneuralnetworksoperatorsurrogatemodelspartialdifferentialequationscardiacsimulationdigitaltwins

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This survey sets out to establish that Scientific Machine Learning (SciML)—the deliberate combination of physics-based mathematical models with data-driven machine learning—is a workable and valuable strategy for problems governed by partial differential equations. It argues that neither approach alone is enough: physics-based models are robust but computationally heavy and need unknown constitutive laws and parameters, while pure data-driven models ignore causality and generalize poorly when data are sparse. The paper builds a unified mathematical description of both sides, then surveys the main hybrid families: surrogate models trained on high-fidelity simulations, physics-informed neural networks that put PDE residuals into the loss, and operator learning that approximates entire input-to-solution maps. The capstone claim is that these methods have been successfully applied to the Integrated Heart Model, a multiphysics simulation of human cardiac function, in seven concrete tasks from parameter estimation to latent-dynamics acceleration. If the survey is right, SciML is the route to patient-specific digital twins and to affordable many-query simulations.

What carries the argument

The load-bearing object is the physics-informed loss function: a neural-network hypothesis space whose training objective combines the standard data-fitting term with residuals of the PDE, boundary and initial conditions, and optionally a regularization term. This is what lets the same architecture serve as forward solver, inverse parameter estimator, or surrogate: one changes only which variables are trainable and which terms enter the loss. Secondary machinery includes operator networks, which learn maps between function spaces rather than pointwise values, and reduced-order or latent-dynamics models that compress high-fidelity cardiac simulations into a few latent coordinates. The paper also leans on a formal analogy between finite-element trial spaces and neural-network hypothesis spaces, which frames the whole survey.

What would settle it

Take one of the seven cardiac tasks, for instance the multifidelity PINN estimation of ionic parameters: run it on a public dataset with known ground-truth parameters and compare the recovered values and their uncertainty to a standard finite-element calibration. If the PINN does not meet or beat the baseline within the reported accuracy, the flagship demonstration of SciML fails. Alternatively, on a high-dimensional (say $d=10$) linear PDE with known solution, measure PINN test error versus number of collocation points; if the error shows no better than exponential-in-dimension scaling, the paper's high-dimensional efficiency premise is false.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that physics and machine learning are not competing paradigms but complementary components of a single modelling strategy. The mechanism is to take the loss function of a neural network and add terms that encode the residual of the governing differential equations, boundary conditions, or data from a high-fidelity solver, so the network is pushed toward physically admissible solutions. The survey classifies the field into three families—surrogate modelling of full-order models, physics-informed learning (PINNs, VPINNs, Deep Ritz), and operator learning (DeepONet, neural operators, neural ODEs)—and then presents seven cardiac applications (ionic parameter estimation by multifidelity PINNs, physics-aware inverse electrocardiography, multiscale learning of microscopic dynamics, time-dependent operator learning for multiphysics coupling, NN surrogates for sensitivity analysis and Bayesian estimation, and Latent Dynamics Networks to accelerate electrophysiology) as evidence that the hybrid approach delivers. The paper does not present new numerical results; it curates and frames existing ones to make the case that SciML is mature enough for flagship biomedical simulation.

Load-bearing premise

The survey's flagship conclusion—that SciML has been successfully applied to simulate the human cardiac function—is inherited from the authors' earlier publications and is presented without independent reproduction, error bars, or comparison to baselines in this paper.

Editorial extensions

If this is right

Cardiac simulations can move toward real-time and patient-specific use, because neural surrogates and latent-dynamics models reduce the cost of repeatedly solving the same multiphysics model.
Inverse problems that are ill-posed in classical terms—like estimating ionic parameters from body-surface potentials—become tractable when the PDE residual is embedded in the loss function.
Operator learning means a model trained on one set of inputs (geometries, material laws, boundary data) can predict solutions for unseen inputs without retraining.
Sensitivity analysis and Bayesian parameter estimation become affordable because a cheap neural-network surrogate replaces thousands of full-order solves.
The same hybrid recipes should transfer to other multiscale, multiphysics problems beyond cardiology, such as fluid-structure interaction or materials design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication the paper leaves implicit is that the taxonomy itself is the contribution: by ordering the field into surrogate, physics-informed, and operator-learning families, the survey shapes which combinations of methods are tried next.
The paper's high-dimensional efficiency claim is worth testing directly: its own approximation bound $N_1 \sim (1/\varepsilon)^{n/s}$ grows exponentially in the input dimension $n$, so a controlled experiment on a $d=10$ PDE would show whether the surveyed successes rely on problem-specific structure rather than general neural-network power.
A testable extension would be to run the seven cardiac tasks on a standardized public benchmark with error bars and baseline comparisons, turning the survey's inherited evidence into independently reproducible results.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a review paper, not a new-results paper. Sections 2 through 4 are accurate restatements of FEM and ML theory with correct citations; Section 5 is a summary of your own cardiac program. The paper will serve as an orientation piece for newcomers to SciML.

What it does well: the mathematics I checked is quoted correctly—Cybenko, Yarotsky, the GKP bounds, and the VC-dimension estimate are all faithful. The PINN exposition names real limitations, including spectral bias and the higher cost compared with FEM for standard forward problems. The surrogate-model cost comparison in Section 4.1 is a genuinely useful rule of thumb. The authors are also honest about the scope: they state that they had to select topics and that their classification is not rigid.

The soft spots are two, and I think the reader's note flags the right ones. First, Section 4.2.1 says PINNs are a compelling alternative for high-dimensional PDEs because neural networks efficiently approximate functions in high dimensions. That sits in direct tension with the paper's own Theorem 3.2 and Theorem 3.5, both of which give exponential dependence on dimension. Unless you add a structural assumption—low intrinsic dimension, analyticity, or an underlying low-dimensional manifold—that claim is not justified. As written, it is the standard PINN marketing phrase, and it is wrong in this paper. Second, the abstract promises a 'successful application' to cardiac simulation, but Section 5.2 through 5.8 are all self-cited summaries of the authors' prior work, without error bars, baselines, or independent reproduction. That does not make the underlying results false, but the survey itself does not demonstrate the success it advertises. A careful reader should treat Section 5 as a pointer to the primary literature, not as evidence.

I agree with the reader's conditional verdict. The paper deserves a serious referee, because it is a competent survey by leading authors with a clear pedagogical value. It is not a research advance, and it should not be cited as one. Who gets value from it: graduate students, applied mathematicians moving into SciML, and anyone needing a map of the field with canonical references. My recommendation is to engage with it as a survey, and if it comes back for revision, push the authors to qualify the high-dimensional claim and to label Section 5 as a summary of their own prior work with pointers. Both are easy fixes, and they would make the paper significantly more defensible.

Referee Report

2 major / 6 minor

Summary. The paper is a broad survey/tutorial on Scientific Machine Learning (SciML). It first reviews digital models and their mathematical/numerical foundations (Section 2), then introduces machine learning concepts, architectures, training algorithms, and approximation theory (Section 3), and then describes SciML paradigms such as surrogate modeling, PINNs, VPINNs, Deep Ritz, operator learning, and hidden-dynamics discovery (Section 4). The final part (Section 5) presents the authors' Integrated Heart Model program and summarizes seven applications of SciML to cardiac simulation, which the abstract describes as the 'successful application of SciML to the simulation of the human cardiac function.' The central thesis is that combining physics-based models with data-driven algorithms can inject physical knowledge into ML while using data to enhance or accelerate physics-based models.

Significance. As a pedagogical survey, the paper has genuine merit: it provides a readable mathematical introduction to neural networks, cites approximation-theoretic results (Cybenko, Yarotsky, G"uring-Kutyniok-Petersen) accurately, distinguishes observational/inductive/learning biases, and gives practical cost criteria for surrogate models, e.g. Eq. (74). It also explicitly acknowledges some limitations of PINNs, such as spectral bias and higher cost than FEM for forward problems. However, the paper's flagship claim—that SciML has been successfully applied to cardiac simulation—is not established in this manuscript: Sections 5.2-5.8 report the authors' own prior results without external benchmarks, error bars, or reproduction. In addition, a load-bearing statement in Section 4.2.1 about the efficiency of neural networks in high-dimensional spaces conflicts with the paper's own approximation-theoretic lower bounds in Section 3.2.5. These issues materially affect the strength of the conclusions and require revision.

major comments (2)

[4.2.1] The paragraph claiming that 'in cases involving PDEs in high-dimensional spaces... PINNs offer a compelling alternative. This is due to the neural networks' ability to efficiently approximate functions in high-dimensional spaces' is not supported by the paper's own approximation theory. Theorem 3.2 (Eq. (32)) gives N1 ≳ ε^{-(n-1)/s} for shallow networks on W^{s,2}((0,1)^n), and Theorem 3.5 gives a similar exponential-in-dimension lower bound for deep ReLU networks. The paper itself states in Section 3.2.5 that 'shallow FFNNs suffer from the curse of dimensionality' and that deep FFNNs 'do not break the curse of dimensionality.' Since Section 4.2.1 states no structural assumptions (such as low intrinsic dimension, analyticity, or special solution classes), the high-dimensional-efficiency claim is internally inconsistent. The authors should either remove the claim or qualify it with a concrete problem class for which positive results exist.
[5.2-5.8] The abstract's claim of 'the successful application of SciML to the simulation of the human cardiac function' is inherited from seven summaries of the authors' own prior publications in Sections 5.2-5.8. The survey provides no error bars, no comparisons against independent baselines or full-order models, and no reproduction or external validation for these results. Because this is the paper's central demonstration, the authors should either include quantitative evidence (e.g., accuracy metrics against FOM solutions or clinical data) or explicitly reframe Section 5 as a programmatic description of the authors' ongoing research rather than an established, independently verified success.

minor comments (6)

[2.2] The word 'shortbreaking' should be 'shortcomings'.
[3.2.1] In the density-estimation bullet, 'unveal' should be 'unveil'.
[3.2.4] The activation function is spelled 'Heavyside' but should be 'Heaviside'.
[3.2.6] In the RMSProp paragraph, 'RSMProp' is a typo for 'RMSProp'.
[3.2.3] The phrase 'euclidian distance' should be 'Euclidean distance'.
[3.3.2] The text uses 'LMM' for 'Large Language Models'; the standard abbreviation is 'LLM'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the paper is an expository survey; its self-cited cardiac illustrations are not predictions derived from the review's own equations.

full rationale

This paper is an expository survey rather than a derivation chain. The educational core (Sections 2 through 4) compiles standard, externally grounded results: Cybenko's universal approximation theorem, Yarotsky's and G"uring--Kutyniok--Petersen complexity bounds, VC-dimension generalization estimates, and standard optimization methods. These are cited to independent sources and are not defined in terms of the paper's own conclusions. The later cardiac-application sections summarize the authors' own prior work, but this is not a circular reduction: the prior papers are external publications containing their own numerical experiments, and the review does not rename a fitted parameter as a prediction or derive an output from an input by construction. The only notable tension is internal consistency, not circularity: Section 4.2.1 claims that PINNs are compelling for high-dimensional PDEs because neural networks efficiently approximate high-dimensional functions, while Theorem 3.2 (Eq. 32) shows the required number of neurons grows exponentially in dimension for fixed regularity. That is a possible overstatement or missing structural assumption, but it is not a case of a result being equivalent to its inputs by definition. Self-citation is present and could raise evidentiary concerns, but under the hard rules it does not constitute circularity unless the load-bearing argument reduces to an unverified self-citation; here the cited prior works are independent publications. Accordingly, no significant circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters: the paper is a survey and fits nothing to data. Numeric quantities it reports (learning rates, Adam hyperparameters beta1=0.9, beta2=0.999, T=10^4 in positional encoding) are standard hyperparameters from the cited literature. No invented entities: the paper postulates no new physical or mathematical entities; its content is a review of existing methods and its own prior program.

assumptions (4)

standard math Classical approximation theorems are valid as stated (Cybenko UAT, Yarotsky, Guring-Kutyniok-Petersen bounds, VC-dimension generalization bound).
Invoked in Section 3.2.5 (Theorems 3.1-3.5, eq. 35) to support the expressivity and generalization discussion.
domain assumption The Integrated Heart Model [75] reliably simulates cardiac electromechanics, valve dynamics, circulation, perfusion, and torso potential.
Section 5.1 bases the flagship application on the IHM as a high-fidelity physics-based model; the paper gives no validation evidence for the IHM itself.
domain assumption The authors' previously published SciML results (Sections 5.2-5.8) are correct and generalizable.
The survey's success claims are supported by self-citation to prior peer-reviewed papers; no independent reproduction or benchmark is provided.
domain assumption Neural networks can efficiently approximate solutions of high-dimensional PDE problems.
Section 4.2.1 claims PINNs 'offer a compelling alternative' in high dimension; this conflicts with the exponential dimension dependence in Theorem 3.2 cited in Section 3.2.5, and the tension is not discussed.

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Cite this review

Pith. "Pith review of Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning." pith.science (2026). https://pith.science/paper/LVZQ25WO

@misc{pith2026250118708,
  author       = {Pith},
  title        = {Pith review of: Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVZQ25WO}},
  note         = {Machine review of arXiv:2501.18708}
}

read the original abstract

Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data, increasingly cheap computing power, and the development of powerful ML algorithms. SciML leverages the physical awareness of physics-based models and the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into ML algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and nonlinear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and ML algorithms and presenting the most popular ML architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by PDEs. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socioeconomic importance that poses numerous challenges on both the mathematical and computational fronts. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

Figures

Figures reproduced from arXiv: 2501.18708 by the authors.

**Figure 1.** The abstract framework. On the other hand, Artificial Intelligence has gained momentum over the last two decades. In particular, we can refer to algorithms empowered by data–trained artificial neural networks. Three triggering factors have contributed to such a rapid development: (i) relatively cheap computing power, offered especially by cloud services, and GPUs which can carry out very fast computations; (ii) the … view at source ↗

**Figure 2.** Problems, solutions, and errors in digital models [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** A simplified example of a decision tree to predict the cardiovascular risk for a diabetic man [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (43 more)

**Figure 4.** Figure 4: A biological neuron (left) and the perceptron (right) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: Activation functions: the Heaviside function (a), ReLU (b), sigmoid (c), and hyperbolic tangent [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: The sigmoid function with different transitions’ steepness and location [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: A Feed Forward Neural Network with L = 3 and the zoom on a neuron at the layer ℓ = 2. Algorithm 1 Feed Forward Neural Network procedure FFNN (xˆ, w) a [0] = xˆ for ℓ = 1, . . . , L do z [ℓ] = W[ℓ]a [ℓ−1] + b [ℓ] a [ℓ] = σ(z [ℓ] ) end for return y = f(xˆ; w) = a [L] end…

**Figure 8.** Figure 8: The ideal model ˆf and the really computed model ˆf ∗ H,S because of the infinite dimension of the space itself, thus a first approximation of (27) consists in looking for a model in a suitable hypothesis space H subset of Y X , that is, computing ˆfH = argmin f∈H R(f)…

**Figure 9.** Figure 9: Underfitting (left), optimal fitting (centre), and overfitting (right) for a classification task [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]

**Figure 10.** Figure 10: The empirical risk (training error), the expected risk, and the generalization error versus the [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]

**Figure 11.** Figure 11: Train and validation errors versus the capacity of the hypothesis space [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]

**Figure 12.** Figure 12: (Left) One update of the descent method with the descent direction given by the opposite [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]

**Figure 13.** Figure 13: E[L(w(k) ) − L(w∗ )] for the SDG method with two different constant choices of the learning rate η. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]

**Figure 14.** Figure 14: Subdivision of the dataset in training, test and validation sets [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]

**Figure 15.** Figure 15: Comparison of training and validation empirical risks for hyperparameters tuning [PITH_FULL_IMAGE:figures/full_fig_p038_15.png]

**Figure 16.** Figure 16: Some of the most common model layers in NNs. Fully connected layers (a), a convolutional [PITH_FULL_IMAGE:figures/full_fig_p040_16.png]

**Figure 17.** Figure 17: Scaled Dot–Product Attention layer (64) on the left, and Multi–Head Attention layer (66)– [PITH_FULL_IMAGE:figures/full_fig_p042_17.png]

**Figure 18.** Figure 18: An example of Convolutional Neural Network [PITH_FULL_IMAGE:figures/full_fig_p042_18.png]

**Figure 19.** Figure 19: The graph of a traditional Recurrent Neural Network [PITH_FULL_IMAGE:figures/full_fig_p043_19.png]

**Figure 20.** Figure 20: Graph Neural Network. Different coloured regions of an image are associated with the nodes [PITH_FULL_IMAGE:figures/full_fig_p045_20.png]

**Figure 21.** Figure 21: Encoder–decoder architecture First, in the input embedding phase, the input sentence is decomposed into n tokens which are transformed into the corresponding word embeddings of fixed length dmodel. The matrix X containing the word embeddings is added to the positiona…

**Figure 22.** Figure 22: The Transformer[243] Algorithm 14 The Multi–Head Self–Attention layer: H = MSA(X) Self–Attention Projection Q = X WfQ, K = X WfK, V = X WfV Multi–Head Attention for i = 1, . . . , nh do Qi = QWQ i , Ki = KW K i , Vi = V WV i Ai = Attention(Qi , Ki , Vi) = softmax Qi…

**Figure 23.** Figure 23: Swin Transformer. (a) The hierarchical partition in the Swin Transformer. Red lines bound [PITH_FULL_IMAGE:figures/full_fig_p049_23.png]

**Figure 24.** Figure 24: Generative Adversarial Network autonomous vehicles, and resource management. Examples of these architectures are Deep Q–Networks (DQN, 2015), Policy Gradient Methods, and Actor–Critic Methods. Hybrid architectures. Finally, hybrid architectures (supervised and unsuper…

**Figure 25.** Figure 25: Number of trainable parameters for notable AI models. GPT-4 (2023) with 10 [PITH_FULL_IMAGE:figures/full_fig_p051_25.png]

**Figure 26.** Figure 26: Training compute of notable models in the number of FLoating Point Operations (FLOP). [PITH_FULL_IMAGE:figures/full_fig_p051_26.png]

**Figure 27.** Figure 27: Cooperation between Digital Models and Machine Learning algorithms [PITH_FULL_IMAGE:figures/full_fig_p054_27.png]

**Figure 28.** Figure 28: Physics–based approaches, how ML algorithms can improve digital models [PITH_FULL_IMAGE:figures/full_fig_p055_28.png]

**Figure 29.** Figure 29: Data–driven based approaches, how digital models can improve ML algorithms [PITH_FULL_IMAGE:figures/full_fig_p056_29.png]

**Figure 30.** Figure 30: The computational domain and setting for problem 75 [PITH_FULL_IMAGE:figures/full_fig_p058_30.png]

**Figure 31.** Figure 31: Two possible FFNNs for the parametric Navier–Stokes equation (75) [PITH_FULL_IMAGE:figures/full_fig_p059_31.png]

**Figure 32.** Figure 32: A computational mesh made of triangles (left), a [PITH_FULL_IMAGE:figures/full_fig_p061_32.png]

**Figure 33.** Figure 33: The collocation nodes (left) and the PINN (right). We follow the branch [PITH_FULL_IMAGE:figures/full_fig_p062_33.png]

**Figure 34.** Figure 34: The minimization process in PINNs where the loss function is L(w) = LPDE(w) + αBC LBC (w) with LPDE(w) = 1 2NPDE N XPDE i=1 [PITH_FULL_IMAGE:figures/full_fig_p062_34.png]

**Figure 35.** Figure 35: Deep Ritz Models While PINNs evaluate residuals only by using automatic differentiation and do not discretize the PDE, ODIL first discretizes the PDE by a grid method (a Full Order Method), e.g., finite differences or finite volumes, so that the residuals of the discr…

**Figure 36.** Figure 36: The NN by Chen and Chen approximating any (non)linear continuous operator [229] [PITH_FULL_IMAGE:figures/full_fig_p071_36.png]

**Figure 37.** Figure 37: A Deep Operator Network (DeepONet) is replaced by a kernel convolution R Ω κ(x, xˆ)a [ℓ−1](ˆx)dxˆ. Notably, in the discrete case (Ωℓ = {1, 2, . . . , Nℓ}), the kernel integral reduces to a matrix multiplication, ensuring consistency between discrete and continuous fo…

**Figure 38.** Figure 38: A Neural Operator u W σ P Layer 1 Layer 2 Layer L Q v (F −1(KF(a [1])))(x) + b(x) a [1] [PITH_FULL_IMAGE:figures/full_fig_p074_38.png]

**Figure 39.** Figure 39: A Fourier Neural Operator of efficient parametrizations are described: graph neural operators, multi–pole graph neural operators, low–rank neural operators, and Fourier neural operators. Fourier Neural Operators (FNO). FNO were proposed in [145]. Instead of working in…

**Figure 40.** Figure 40: (a) The scalable Operator Transformer scOT that is the backbone of Poseidon, (b) one of the [PITH_FULL_IMAGE:figures/full_fig_p084_40.png]

**Figure 41.** Figure 41: The neural network architecture of ICON In [259], the idea of in–context learning has been extended to learn operators that underlie differential equations. An In–Context Operator Newtowrks ICON model is a transformer encoder–decoder architecture [243] (see Sect. 3.3…

**Figure 42.** Figure 42: The heart, the cardiac muscle’s cells (cardiomyocytes), and a sarcomere, the fundamental [PITH_FULL_IMAGE:figures/full_fig_p088_42.png]

**Figure 43.** Figure 43: Interactions among fundamental processes occurring in the cardiac function [PITH_FULL_IMAGE:figures/full_fig_p089_43.png]

**Figure 44.** Figure 44: On the left, the computational domain of the inverse problem of electrocar [PITH_FULL_IMAGE:figures/full_fig_p096_44.png]

**Figure 45.** Figure 45: Using model learning with latent variables to reduce the computational cost associated with [PITH_FULL_IMAGE:figures/full_fig_p100_45.png]

**Figure 46.** Figure 46: Output of the Bayesian estimation presented in Sec. 5.7. We depict the posterior distribution [PITH_FULL_IMAGE:figures/full_fig_p106_46.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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