Open research questions in Model Reduction and Neural Networks

What the literature leaves open

• Future research will focus on extending this methodology to other situations where evaluat- ing the high fidelity kinetic model is significantly demanding when high-dimensional uncertainties are present, like diffusion limits [1] and collisional plasmas [23] charac- terized by the Landau operator.

  Structure- and Asymptotic-Preserving Deep Neural Surrogates for Uncertainty Quantification in Multiscale Kinetic Equations · 2026 · DOI

• Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.

  FlashPDE: A Drop-In Fused Triton Operator Library for Neural PDE Solvers · 2026

• Future work should focus on extending the approach to more complex surface evolution mod- els, incorporating uncertainty estimates on the pre- diction, and applying the method to experimental surface data from ion-beam or plasma–wall interac- tion studies. The model is trained exclusively on sim- ulated data generated from a fixed form of the KS equation, and its performance on experimental sur- faces remains to be demonstrated. Several limitations of the present study should be noted.

  Vision transformer based parameter estimation for 2D Kuramoto–Sivashinsky models using physics informed features · 2026 · DOI

• The MIPINN methodology exhibits enhanced robustness, reduced parameter uncertainty, and stable convergence behavior under sparse data constraints even in three-dimensional multiparameter inverse heat conduction scenarios.

  Modified-inverse physics informed neural networks for determination of orthotropic thermal conductivities · 2026 · DOI

• Experimental techniques, such as 4D flow MRI, PIV, or Doppler ultrasound, often yield data that are sparse, noisy, or under-resolved, particularly near vessel walls and in regions of complex flow.

  Physics-Informed Neural Networks for the High-Resolution Reconstruction of Flow Measurement Indicators in Fluid Dynamics · 2026

• Across five PDE regimes, the results show that diffusion-based learned discretization is competitive with adaptive-mesh and reduced-order baselines, with particularly strong gains in regimes where fixed or handcrafted allocation is insufficient.

  Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance · 2026

• From Classical Physics to Scientific Machine Learning Scientific modelling throughout history is characterized by attempts to increase understanding, prediction and simulation of nature. Since the emergence of deterministic laws in classical mechanics, the concept of physical phenomena modelling based on governing equations of physics has been formed. Later, the area was broadened significantly as a result of new field theories, statistical mechanics, and computational science. However, when problems became even more nonlinear, high-dimensional, and complex for computational analysis, pure approaches based on first principles have become less efficient due to their inability to model turbulence, biochemistry, climate and other complicated phenomena. The rapid growth of machine learning and artificial intelligence led to a major shift and the introduction of modelling approaches based on pattern recognition from data rather than physical laws. These approaches enabled the approximation of nonlinear systems, acceleration of simulations, and discovery of hidden relationships within scientific datasets. However, pure data-driven models had low physical interpretability, robustness and consistency with scientific laws. Thus, a need has emerged for approaches that combine physical knowledge with machine learning. Hybrid modelling or scientific machine learning combines physical and learning approaches in one computational system. It has seen several successful developments, among which physics-informed neural networks (PINN), operator learning, surrogate models, and physics-based learning are to be highlighted. All of the mentioned methods unite the advantages of both approaches by combining machine learning predictive power with the reliability and physical interpretability of scientific laws. Therefore, scientific modelling has started to move away from purely equation-based computational techniques to adaptive hybrid models combining theory, simulation, data and learning in one framework. 10.2 The Emerging Scientific Paradigm The integration of artificial intelligence and physical modelling is not only an improvement of technology but also a more profound transformation of scientific research techniques in general. Historically, science mainly relied on two major approaches – theoretical modelling and experimentation. Later, simulation became the third technique that was widely used for science purposes. Today, machine learning is increasingly emerging as a fourth pillar of scientific investigation alongside theory, experimentation, and computational simulation. Machine learning does not replace physics in its function, but rather serves as a computational tool that can speed up simulations, approximate unknown relations, deal with large datasets, and help conduct scientific reasoning.

  From Newton to Neural Networks: A Review of Data-Driven Physical Modelling and the Rise of Physics-Informed AI · 2026 · DOI

• So, to address these gaps in this study we have established a very limited data guided multi-stage inverse physics-informed neural network (PINN) architecture integrated with SCWR thermal-hydraulic model THRUST that simultaneously recovers transient bulk fluid temperature and transient wall heat flux distributions under the assumption of constant mass flux and a prescribed range of linear pressure drop.

  A Sparse-Data-Guided Inverse PINN for the Recovery of Thermal-Hydraulic Fields in Supercritical Water Reactor · 2026 · DOI

• All benchmarks in this study are two-dimensional; whether the design-principle ranking established here transfers to three-dimensional interface transport, where collocation requirements grow substantially with dimension, remains to be verified. Whether such schemes can recover the benchmark-specific optima found here, particularly the four-order-of-magnitude shift in optimal weik between rigid-body and strongly deforming flows, is an open question and a natural next step toward eliminating manual eikonal tuning. 3 Comparison with Classical Methods and Prior PINN Work has not been reported previously; while Mullins et al.

  A systematic study of physics-informed neural networks for the level-set interface advection · 2026 · DOI

• Compared with standard and regularized MFA, PI-MFA produces more physically faithful reconstructions and, for physically inconsistent data, lower approximation errors, while offering computational advantages over tested physics-informed neural networks.

  A Physics-Informed B-Spline Framework for Continuous Approximation of Flow Data · 2026

• The study focuses on reducing the field discretization error through the coupling of extrapolation techniques with SubGrid strategies, an aspect rarely explored in the literature.

  Richardson extrapolation with SubGrid to reduce the field discretization error in computational fluid dynamics · 2026 · DOI

• In this work, we propose P2FGAN, a pressure-conditioned generative adversarial network that synthesizes high-fidel- ity solid-propellant combustion temperature-field images under continuous pressure operating conditions, providing a fast surrogate to replace repeated CFD solves. Compared with prior conditional GANs that inject a scalar condition only once, P2FGAN continuously enforces pressure consis- tency during generation via Fourier-embedded conditioning and multi-scale modulation, yielding more stable structures and better controllability across the full pressure range. Experiments on Fluent-generated datasets demonstrate that P2FGAN improves image fidelity and condition alignment over representative baselines while achieving millisecond- level inference latency, suggesting an overall speedup on the order of 105~106 over CFD for multi-condition visualiza- tion. This efficiency is obtained by trading one-time offline training and model storage for online synthesis, and future work will extend the framework with additional physics constraints and broader operating parameters, building on the central insight that continuous-condition injection can couple scalar operating states with the evolution of multi- scale field structures. Although the experimental validation in this study is conducted under a single combustion configuration to establish the pressure-conditioned temperature-field image synthesis framework, the proposed method is not inherently tied to the specific configuration used in this study. The core mechanism of P2FGAN treats pressure as a continu- ous control signal and maintains its influence throughout the generation process via a cross-scale condition communica- tion pipeline, ensuring pressure consistency in the gener- ated results. As long as the task remains temperature-field image synthesis driven by scalar operating conditions, this mechanism is expected to be transferable to other combus- tion setups. For other configurations or conditions, such as swirl-stabilized flames, bluff-body stabilized flames, con- fined combustors, or solid-propellant setups with different grain or port geometries, the framework can be extended by retaining the generator–discriminator backbone while introducing configuration-related information as additional conditioning inputs. Specifically, geometric or flow-related parameters, or a low-dimensional geometry code, can be incorporated with pressure into the conditioning pathway, and the model can be trained on multi-configuration data- sets. Cross-configuration generalization can be evaluated using a leave-one-configuration-out protocol. Future work will focus on multi-condition and multi-configuration mod- eling, integrating physics-aware regularization, and extend- ing the framework to broader operating parameters, aiming to further strengthen the coupling between scalar operating states and the evolution of multi-scale field structures. Author contributions Haiyang Zhao developed the proposed P2FGAN model, conducted experiments, and wrote the initial draft.Haoran Yu assisted in model implementation and experimental verification.Baodi Liu contributed to data preprocessing and visualization.Zhenyu Zhang participated in performance evaluation and result analysis.Xiang Lyu contributed to dataset preparation and result validation.Weifeng Liu provided supervision, conceptual guidance, and critical revisions.All authors reviewed and approved the final version of the manuscript. Funding Funding was provided by National Key Laboratory of Solid Rocket Propulsion (Grant SY41YYF202403051). Data availability No datasets were generated or analysed during the current study.

  A pressure-conditioned generative adversarial network for efficient temperature field visualization in combustion simulations · 2026 · DOI

• Although JacobiNet enables geometry editing operations with differentiable Jacobians, thereby expanding the range of geometries amenable to coordinate transformations, mappings for highly complex domains remain unknown. Generating supervised paired data for highly complex geometries, as discussed in Section 4, such as branching vessels, remains an open challenge due to the difficulty of constructing consistent point-wise correspondences.

  Solved in unit domain: JacobiNet for differentiable coordinate-transformed PINNs · 2026 · DOI

• Future work may focus on exploring optimal sensor placement for data reconstruction methods and the development of parallel computing implementations to further reduce computation time, enabling the processing of even larger datasets and expanding the utility of the framework in complex fluid dynamics applications.

  Ensemble Kalman filter for data assimilation coupled with low-resolution computations techniques applied in fluid dynamics · 2026 · DOI

• In this paper, we have developed and analyzed a Bi-fidelity Asymptotic-Preserving Neural Network (BI-APNNs) framework for solving forward and inverse problems for the semiconductor Boltzmann equation. This work builds upon the standard APNNs methodology, specifically addressing a key practical bottleneck: the slow convergence. The slower convergence of the standard APNNs can be observed in both the training loss and the relative ℓ2 error curves, particularly in the computa- tionally challenging fluid-dynamic regime. In contrast, the BI-APNNs shows faster overall convergence of the macroscopic density, ρ, in the computationally challenging fluid-dynamic regime. The core of our approach is a novel bi-fidelity decomposition of the density. In the forward problem, it allows a pre-trained low-cost network to capture the dominant physical behavior, while a much smaller network learns the re- maining high-fidelity correction. Through a series of numerical experiments, we have demonstrated the key advantages of this bi-fidelity formulation. The results confirm that, particularly in the small-ε regime, the BI-APNNs not only significantly accel- erate training convergence but also improve the accuracy of the forward problem solution. Most importantly, we have shown that this enhanced accuracy translates directly into superior performance for the associated inverse problems when com- pared to the standard APNNs. This work underscores the potential of bi-fidelity methods to enhance physics-informed neural networks, making them a more efficient and reliable tool for complex multiscale kinetic problems. Several promising directions for future research emerge from this work. On the theoretical front, our convergence analysis could be further refined by incor- porating tools from Barron spaces [12] and developing posterior error estimates to more precisely quantify the network’s approximation capabilities. From a practical standpoint, extending the BI-APNNs framework to higher-dimensional problems is a key next step. Furthermore, the computational efficiency demonstrated by the BI-APNNs makes them an particularly attractive candidate for uncertainty quantifi- cation (UQ) problems, where the need for numerous forward solves often creates a computational bottleneck.

  A bi-fidelity based asymptotic-preserving neural network for the semiconductor Boltzmann equation and its inverse problem · 2026 · DOI

• Engineering with Computers (2026) 42:104 1 3 104 Page 24 of 24 Future work will focus on enhancing HDD robustness and generality through improved interface coupling, itera- tive schemes, and bidirectional adaptivity, allowing tran- sitions from reduced back to full models when necessary.

  Error estimates of upscaled poroelastic models in thin domains for hybrid modeling via heterogeneous domain decomposition · 2026 · DOI

• The wake observation region Ω2 for the objective function is fixed at x ∈ [1, 4], y ∈ [0, 2]; the sensitivity of model performance to the choice of observation domain size, location, and exclusion of near-wall regions across varying Reynolds numbers has not been systematically analyzed.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The discrete adjoint framework from DAFoam differentiates through the fully converged steady RANS system; the computational cost, memory requirements, and scalability of this adjoint-based training approach for larger domains, unsteady flows, or three-dimensional configurations in practical CFD solvers have not been characterized.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The Reynolds-force model was trained and validated exclusively on cylinder flow data at Re ≈ 300−300,000 with the near-wall region remaining laminar; the generalization of the trained neural network closure to separated flows around bluff bodies with different geometries or highly turbulent near-wall regions has not been demonstrated.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The hybrid implicit/explicit coupling strategy uses the baseline Spalart–Allmaras model to provide baseline eddy viscosity while the neural network predicts residual Reynolds-force corrections; the applicability and performance of this hybrid approach with alternative baseline turbulence models (k-ε, k-ω, RSM) remains unexplored.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The neural network inputs are normalized using the bounded transformation η̂ = η/(1 + |η|) to ensure numerical robustness; the impact of alternative normalization schemes or input feature scaling strategies on the stability and accuracy of the coupled NN-RANS solver during iterative solving has not been evaluated.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The objective function (equation 5.7) weights the velocity and Reynolds-force discrepancies with λ = 0.1; the paper does not explore how different weighting schemes or domain-specific weight distributions (varying λ across Ω1 and Ω2) affect the convergence and generalization of the neural network turbulence model across different Reynolds numbers.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• The wall-damping function for the Reynolds-force neural network is currently defined using a cubic polynomial with a fixed threshold d* = 0.1D; the sensitivity of turbulence closure predictions to alternative damping function formulations and threshold values across the Reynolds number range Re ≈ 300−300,000 has not been systematically investigated.

  Turbulence closure in Reynolds-averaged Navier–Stokes and flow inference around a cylinder using physics-informed neural networks and sparse experimental data · 2026 · DOI

• This paper mainly reviews the algorithms of AI for PDEs (including Physics-Informed Neural Networks, Operator Learning, and Physics-Informed Neural Operators), the related theoretical research, and applications in the forward and inverse problems of computational mechanics, including solid mechanics, fluid mechanics, and biomechanics. Based on the current state of research, possible future directions for AI for PDEs in computational mechanics might include: (1) Nonlinear Problems: The core component of AI for PDEs is neural networks, which have strong nonlinear capabilities. Therefore, future research on nonlinear problems theoretically has good prospects. For linear elasticity problems in solid mechanics, finite element methods are already quite perfect due to the positive definiteness and sparsity of the stiffness matrix, which can be quickly and accurately solved by direct matrix inversion. However, for some nonlinear problems, traditional finite element methods either solve the nonlinear equation system directly using Newton’s iterative method or transform and solve it explicitly using incremental steps. Neural networks, due to their inherent nonlinear capabilities, theoretically have an advantage in solving nonlinear problems in computational mechanics. Also, by training operator neural networks with existing numerical simulations or experimental results, and then using operator learning to provide a good initial solution for the initial iteration vector of the nonlinear equation system, the computational efficiency could be greatly improved theoretically. Hyperelastic problems, for example, are a good entry point because they are pathindependent and can be directly formulated as a nonlinear equation system. (2) Complex Phenomena with Inadequate Understanding: For such problems, due to their complexity, the mathematical PDEs descriptions of these issues can only be approximations, meaning the simulation results still differ from actual experimental results. In this case, we can rely on data to fine-tune the results. That is, an approximate solution is first provided by the boundary conditions and an approximate physical equation, and then the simulation results are fine-tuned according to the experimental data, blending a small amount of data with an approximate physical equation, especially for simulating complex phenomena. As humanity’s understanding of the phenomenon becomes clearer, only the physical equations need to be corrected, and this framework remains unchanged. (3) Constitutive Equations: Constitutive equations have always been a core issue in mechanics, where most of the work involves fitting. Typically, experts first construct a specific form of the constitutive model based on some basic physical principles and then fit the parameters in the model according to experimental stress-strain points. However, because of the fitting characteristics of neural networks, theoretically, they can replace constitutive equations.

  Artificial Intelligence For Partial Differential Equations In Computational Mechanics: A Review · 2026 · DOI

• energy-based regularization, or a transition toward intrusive formulations that explic- itly embed the governing equations into the learning process, avenues that are currently beyond the scope of this work. Similarly, another open question is whether one can discover the decoupling of the parameter space from data, i.

  Deep orthogonal decomposition: a continuously adaptive neural network approach to model order reduction of parametrized partial differential equations · 2026 · DOI

Most-cited papers in Model Reduction and Neural Networks

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• Residual-based attention in physics-informed neural networks · Computer Methods in Applied Mechanics and Engineering · 2024 · 155 citations
• Can physics-informed neural networks beat the finite element method? · IMA Journal of Applied Mathematics · 2024 · 154 citations
• Tackling the curse of dimensionality with physics-informed neural networks · Neural Networks · 2024 · 149 citations

Most recent work

• Artificial Intelligence For Partial Differential Equations In Computational Mechanics: A Review · Applied Mechanics Reviews · 2026
• Neural tangent kernel analysis to probe convergence in physics-informed neural solvers: PIKANs vs. PINNs · Computers & Mathematics with Applications · 2026
• Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers · Computational Methods in Applied Mathematics · 2026
• Enhanced Solution for the Advection–Diffusion–Reaction Equation Using the Physics-Informed Neural Network Technique · Mathematics · 2026
• Can We Extract Physics-like Energies from Generative Protein Diffusion Models? · bioRxiv · 2026
• A Novel Fourier Feature Physics‐Informed Neural Networks Based on the Boundary Element Method for Solving Scattering of SH Wave Induced by Complex Topography · Earthquake Engineering & Structural Dynamics · 2026
• A synergistic multi-scale physics-informed neural network for wake flow reconstruction from sparse data · Ocean Engineering · 2026
• A Practical Identifiability Criterion Leveraging Weak-Form Parameter Estimation · Bulletin of Mathematical Biology · 2026
• A Vorticity-Enhanced Physics-Informed Neural Network with Logarithmic Reynolds Embedding · Fluids · 2026
• Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis · Constructive Approximation · 2026

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