Open research questions in Model Reduction and Neural Networks
408 unresolved questions extracted from the limitations and future-work sections of 723 Model Reduction and Neural Networks papers in our library. Each links back to the study that raised it.
What the literature leaves open
The challenge of solving nonlinear, high-dimensional problems with classical numerical methods. The need to balance accuracy and computational cost in PINNs. The limitation of HAM to a local series expansion, which can lead to increased error outside the original domain.
A Unified Approach with Physics-Informed Neural Networks (PINNs) and the Homotopy Analysis Method (HAM) for Precise Approximate Solutions to Nonlinear PDEs: A Study of Burgers, Huxley, Fisher and Their Coupled Form · 2026 · DOIFurther study on the application of PINNs and HAM to other nonlinear PDEs, - Investigation of the effects of domain truncation on the performance of PINNs and HAM
A Unified Approach with Physics-Informed Neural Networks (PINNs) and the Homotopy Analysis Method (HAM) for Precise Approximate Solutions to Nonlinear PDEs: A Study of Burgers, Huxley, Fisher and Their Coupled Form · 2026 · DOIonboard computational constraints, - limited processing power, memory, and energy, - strict control loop requirements (typically 100-1000 Hz)
A Mathematical Review of Reduced Aeroelastic Models, Multiagent Dynamics, and Control Allocation in UAV Systems · 2026 · DOIconvergence of reduced-order models, uncertainty quantification, and machine learning, - efficient prediction of complex multi-physics systems under nonlinear and real-time conditions
A Mathematical Review of Reduced Aeroelastic Models, Multiagent Dynamics, and Control Allocation in UAV Systems · 2026 · DOIFuture work will explore nonlinear extensions of the SLSE for improved small-scale reconstruction, further ablation studies on the CTA ordering and the MTFC fusion weight α, and validation using higher-Reynolds-number datasets. The present results are obtained at a single Reynolds number and a single reference-plane height, and the generalization across different Reτ and reference-plane locations remains to be verified. Huang, “The prediction of external flow field and hy- drodynamic force with limited data using deep neural network,” Journal of Hydrodynamics 35, 549 (2023).
Long-horizon prediction of three-dimensional wall-bounded turbulence with CTA-Swin-UNet and resolvent analysis · 2026In this paper, we proposed CGMPINN, a curriculum-guided physics-informed neural network framework that integrates Gaussian mixture modeling with dynamic curriculum learn- ing for solving partial differential equations. The core idea is to employ a GMM to statistically characterize the dis- tribution of PDE residuals, thereby providing a principled, data-driven quantification of local learning difficulty. A smooth curriculum schedule then progressively transitions training focus from easy, well-conditioned regions to hard, uncertain ones, while a precision-based variance modula- tion mechanism further prioritizes reliable clusters during early training. This dual curriculum is unified through a shared curriculum parameter and optionally coupled with ReLoBRaLo-based self-adaptive loss balancing. System- atic evaluations on six benchmark PDEs demonstrate that CGMPINN consistently achieves the lowest e2 and e∞ errors among all compared methods, with reductions of up to 97.8% in e2 relative to the canonical PINN. No- tably, these accuracy gains incur no significant computa- tional overhead. On the theoretical front, we established 13 From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models formal guarantees for the proposed framework, including uniform equivalence between the curriculum-weighted and standard PDE losses, sublinear convergence of the gradient norm, and a generalization bound with an explicit weighting- induced bias characterization, thereby providing principled justification for the curriculum-guided reweighting strat- egy. An ablation study further confirms that the two core components—GMM-based difficulty quantification and cur- riculum scheduling—are complementary: neither compo- nent alone matches the performance of their combination, and omitting the curriculum schedule can lead to optimiza- tion failure on challenging problems. Despite the encouraging results, several directions merit fur- ther investigation. The current experiments are restricted to one- and two-dimensional domains; extending CGMPINN to high-dimensional PDEs and assessing the scalability of GMM-based difficulty quantification in such settings is an important next step. Developing principled strategies for au- tomatic selection of the number of mixture components K and the update frequency kupd, for instance via information- theoretic criteria, would further reduce manual tuning. Com- bining the curriculum-guided mechanism with domain de- composition approaches and extending the framework to inverse problems with sparse, noisy observations are also natural generalizations. On the theoretical side, the present analysis establishes convergence, loss equivalence, and gen- eralization guarantees under idealized full-batch gradient dynamics; extending these results to a complete iterate-level convergence theory for the practical Adam→L-BFGS train- ing scheme and establishing tighter bounds that formally certify the advantage of dynamic curriculum reweighting over standard PINNs remain valuable open directions.
From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models · 2026Future recommendations: Future research will focus on improving the scalability, robustness, and computational efficiency of the DPIKAN-TO framework.
A Dual Physics-Informed Kolmogorov-Arnold Neural Network Framework for Continuum Topology Optimization · 2026The problem formulation exhibits high-dimensional properties, which are often difficult to handle for data-driven ML methods. The computational cost of PR-DNS limits its use to small-scale systems. The current approach is limited to spherical particles.
Drag modelling for flows through assemblies of spherical particles with machine learning: A comparison of approaches · 2026 · DOIInvestigating the application of the GNN to more complex particle shapes. Exploring the use of other machine learning algorithms to improve the prediction of drag forces. Extending the current approach to larger-scale systems.
Drag modelling for flows through assemblies of spherical particles with machine learning: A comparison of approaches · 2026 · DOIThe performance of hybrid preconditioned iterative solvers on arbitrary unstructured domains is still an open problem. Defining the preconditioner is often challenging when the domain has a non-smooth boundary or includes non-convex regions.
Hybrid Iterative Solvers With Geometry‐Aware Neural Preconditioners for Parametric PDEs · 2026 · DOIAcross this set, PINNs are evaluated on synthetic data generated from analytical solutions; none applies PINNs to advection-diffusion-reaction problems with real experimental or observational data, where model discrepancy, measurement noise, and unresolved physics may violate the assumption that the PDE perfectly governs the system.
Enhanced Solution for the Advection–Diffusion–Reaction Equation Using the Physics-Informed Neural Network Technique · 2026 · DOINone of these studies compares PINN training complexity and convergence behavior on advection-diffusion-reaction equations against the stated O(N_f × P) scaling; the computational cost challenge is identified but no work provides empirical validation of scaling laws or proposes mitigation strategies specific to transport phenomena with competing advection and diffusion timescales.
Enhanced Solution for the Advection–Diffusion–Reaction Equation Using the Physics-Informed Neural Network Technique · 2026 · DOIHuang, “The prediction of external flow field and hydrodynamic force with limited data using deep neural network,” Journal of Hydrodynamics, vol.
Geometry-Aware Surrogate for Real-Time Hydrodynamics Estimation of Autonomous Ground Vehicles in Amphibious Environments · 2026The conclusion is not that one paradigm supersedes the other, but that their principled integration, grounded in the complementary strengths each provides, offers the most credible path toward computationally tractable solutions to the most consequential open problems that neither can address alone.
Partial differential equations in the age of machine learning: a critical synthesis of classical, machine learning, and hybrid methods · 2026 · DOIWe illustrate its efficiency on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.
Sparse data-driven approaches enable the approximation of governing laws of physical processes with parsimonious equations.
Across this set, multilevel domain decomposition methods are evaluated only on high-frequency and multi-scale solutions; none of these studies applies multilevel FBPINNs to advection-diffusion-reaction systems or transport phenomena where multi-scale behavior is driven by competing physical processes rather than oscillatory solutions.
Multilevel domain decomposition-based architectures for physics-informed neural networks · 2024 · DOITraditional GSA methods have limitations when dealing with correlated parameters, - The proposed method addresses this gap by reframing sensitivity analysis as a post-calibration task on Bayesian posterior distributions.
Bayesian-calibrated global sensitivity analysis for mathematical models using generative AI · 2026 · DOIThe neural network architecture selection (two hidden layers with 32 neurons) and training sample size (N1 = 6,000) were chosen based on empirical performance for low-dimensional problems, but systematic guidelines for architecture selection in higher dimensions are lacking.
Bayesian-calibrated global sensitivity analysis for mathematical models using generative AI · 2026 · DOIThe approach is limited to the one-electron case. The approach may not be suitable for very large systems or when the dynamics are highly nonlinear and stiff.
To extend the approach to more complex quantum systems. To improve the scalability and convergence of the approach. To apply the approach to other fields such as computational chemistry and materials science.
Traditional approaches require post hoc normalization of linear updates, which may accumulate drift over long sequences. Existing methods often focus on probabilistic inference requiring sampling procedures, assume specific parametric forms, or model differences rather than states directly.
The choice of preconditioner depends on the specific problem, and practitioners often rely on a combination of theoretical understanding and numerical experimentation. The training of GNNs can be unstable, resulting in preconditioners with weaker effect on the spectrum. The approach needs to handle parametric PDEs with contrast coefficients.
Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers · 2026 · DOIThe paper does not provide a comprehensive comparison with all existing preconditioners. The approach is limited to sparse linear systems. The training of GNNs can be unstable, resulting in preconditioners with weaker effect on the spectrum.
Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers · 2026 · DOIThe paper identifies a gap in the existing literature on surrogate modeling, particularly in the context of complex parametric systems. It highlights the need for a review of established methodologies and new perspectives on surrogate modeling.
Surrogates for Physics-Based and Data-Driven Modelling of Parametric Systems: Review and New Perspectives · 2026 · DOI
Most-cited papers in Model Reduction and Neural Networks
- Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems · ACM Computing Surveys · 2022 · 659 citations
- Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems · Computer Methods in Applied Mechanics and Engineering · 2022 · 654 citations
- Applications of Physics-Informed Neural Networks in Power Systems - A Review · IEEE Transactions on Power Systems · 2022 · 597 citations
- A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials · Computer Methods in Applied Mechanics and Engineering · 2022 · 392 citations
- Analyses of internal structures and defects in materials using physics-informed neural networks · Science Advances · 2022 · 348 citations
- Applications of Deep Learning to Ocean Data Inference and Subgrid Parameterization · Journal of Advances in Modeling Earth Systems · 2019 · 304 citations
- Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems · Computer Methods in Applied Mechanics and Engineering · 2022 · 274 citations
- Respecting causality for training physics-informed neural networks · Computer Methods in Applied Mechanics and Engineering · 2024 · 271 citations
- A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations · Computer Methods in Applied Mechanics and Engineering · 2022 · 269 citations
- Physics-informed Neural Networks (PINN) for computational solid mechanics: Numerical frameworks and applications · Thin-Walled Structures · 2024 · 252 citations
Most recent work
- Discovering interpretable structural dynamic response using physics-informed Kolmogorov-Arnold network · Engineering Applications of Artificial Intelligence · 2026
- Artificial Intelligence For Partial Differential Equations In Computational Mechanics: A Review · Applied Mechanics Reviews · 2026
- Mixed conditional generative adversarial networks for high-dimensional aircraft aerodynamic stealth shape optimization · Engineering Applications of Artificial Intelligence · 2026
- Implementing Non-Abelian Hatano-Nelson Model in Electric Circuits · Physical Review Letters · 2026
- Implementing physics-informed neural networks with deep learning for differential equations · Frontiers in Artificial Intelligence · 2026
- Neural tangent kernel analysis to probe convergence in physics-informed neural solvers: PIKANs vs. PINNs · Computers & Mathematics with Applications · 2026
- Generative models of cell dynamics: from Neural ODEs to flow matching · Communications Biology · 2026
- Online Learning in Idealized Ocean Gyres · Journal of Advances in Modeling Earth Systems · 2026
- Assessment of ChatGPT for Engineering Statics Analysis · Computer Applications in Engineering Education · 2026
- Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers · Computational Methods in Applied Mathematics · 2026
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