Open research questions in Model Reduction and Neural Networks
704 gap statements mined from Model Reduction and Neural Networks papers in our 7.9M-paper local library, which holds 5,181 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
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What the literature leaves open
There is a need for faster surrogates that do not sacrifice physical fidelity. The classical explicit Euler integration scheme may diverge numerically in certain scenarios.
A Physics-Informed, Lipschitz-Certified Neural Correction Framework for Numerically Unstable Explicit Euler Simulation of Synchronous Generator Dynamics · 2026 · DOIThe need for a model that can automatically choose the functional form and fit parameters to experimental data. The lack of a systematic approach to selecting the functional form of the constitutive law. The challenge of ensuring ellipticity in the inputs and convexity in the outputs.
An input-convex physics-informed constitutive artificial neural network for finite hyperelasticity: architecture, approximation scope, and variational identifiability · 2026 · DOIThe need for a generalizable foundation for observing and forecasting non-equilibrium boundary dynamics across various fields. The lack of a unified framework for human population dynamics that accounts for latent boundary conditions.
Local sensitivity analysis shows that the individual Prony parameters are severely ill-conditioned. Parameter identifiability is a challenge, particularly for the Prony series. The approach requires careful selection of hyperparameters and training protocols. The paper identifies the need for further research on the limitations and challenges of the approach.
Learning Memory Operators with Physics-Informed Neural Networks: State Recovery, Kernel Reconstruction, and Parameter Identifiability · 2026 · DOIThe analysis is limited to four kernel parametrizations. The results may not generalize to other kernel families or more complex problems.
Learning Memory Operators with Physics-Informed Neural Networks: State Recovery, Kernel Reconstruction, and Parameter Identifiability · 2026 · DOIClassical representations of the solution, such as POD, have limitations and cannot be used to approximate accurately the solution for varying parameters. Standard MOR approaches are inadequate for solving parametrized problems, particularly in seismic inversion applications. There is a need for efficient solution methodologies that can handle multiple parameters and repeated solutions.
MOR-TL: A novel model order reduction method for parametrized problems with application to seismic wave propagation · 2026 · DOIThe complexities of the mooring system can lead to slight disparities between experimental and simulated results. Spectral leakage can occur when using Fourier-based methods for short- or medium-term observations. The need to isolate the contribution of individual variables to the observed phenomenon in field-recorded data.
The lack of accurate long-range forecasting of sea surface temperature fields. The need for a forecasting framework that can capture the combined influence of recurrent seasonal behavior and shorter-term oceanographic processes.
Combining Multivariate Singular Spectrum Analysis and Dynamic Mode Decomposition for Daily Sea Surface Temperature Forecasting up to 60 Days Ahead in the Marmara Sea · 2026 · DOILack of a probabilistic framework for irreversible kinetics. Need to quantify epistemic uncertainty over admissible histories. Difference between causal and non-causal histories is not well understood.
To ensure accuracy and robustness, a residual-based safeguard is introduced to restore the original physics-based model when neural network predictions are insufficient.
Accelerating nonlinear finite element analysis via residual-aware neural network constitutive models · 2025 · DOIBy tackling the shortcomings of conventional numerical methods in managing fractional derivatives and nonlinearities, this research addresses a significant gap in the literature regarding efficient solution strategies for complex diffusion processes.
Fisher’s information matrix approach for Fourier features physics-informed neural networks for two-dimensional local time-fractional anomalous diffusion equations with nonlinear thermal diffusivity · 2025 · 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 lack of a robust and efficient approach for assessing practical identifiability. The need for a new criterion for evaluating practical identifiability.
To explore the application of the WENDy method to other fields.
Investigating the application of the proposed method to other types of flow images. Exploring the use of different neural network architectures for the quasi-conformal mapping. Studying the effect of varying the levels of noise on the performance of the method.
Coupled Reconstruction of 2D Blood Flow and Vessel Geometry from Noisy Images via Physics-Informed Neural Networks and Quasi-conformal Mapping · 2026 · DOIThe challenges in modelling dislocation interactions, dislocation multiplication or numerical feasibility, - The need for large-scale application of Koopman methods to DDD data, - The current work is based on simulated…
Investigating the application of Koopman methods to larger-scale DDD data, - Exploring the use of physically informed formulations to enhance extrapolation capabilities, - Addressing the challenges in modelling dislocation interactions…
Extension to an I5-dependent network, - Certification of ellipticity for a wider range of deformation…
An input-convex physics-informed constitutive artificial neural network for finite hyperelasticity: architecture, approximation scope, and variational identifiability · 2026 · DOIApplying the framework to other countries or regional systems, - Augmenting the transport equation with a spatial advection or diffusion term to model migration-augmented transport, - Coupling the latent constitutive laws to other socio-economic…
Applying MOR-TL to other complex problems, - Investigating the use of MOR-TL in other fields such as heat transfer and climate…
MOR-TL: A novel model order reduction method for parametrized problems with application to seismic wave propagation · 2026 · DOIEvaluating the method for real-world applications, - Incorporating additional data from ensembles into the training set to increase the likelihood of capturing diverse scenarios, - Extending the method to other areas…
Finally, we discuss open challenges and future directions at the interface of operator theory, approximation theory, and nonlinear control.
As we built directly on the validated benchmark established in this work, several promising directions emerge: Application to Multi-Physics Problems Without Exact Solutions: Having quantified the accuracy and reliability of PINNs against problems with known solutions, the natural next step is to apply them to challenging multi-physics problems— such as coupled heat and mass transfer in complex geometries…
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 · DOIThis article presented an overview of established techniques to construct (parametric) functional surrogates of complex engineering systems using physics-based and data-driven approaches.
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
- Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators · Nature Machine Intelligence · 2021 · 3,359 citations
- DeepXDE: A Deep Learning Library for Solving Differential Equations · SIAM Review · 2021 · 2,342 citations
- An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications · Computer Methods in Applied Mechanics and Engineering · 2020 · 2,096 citations
- Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations · Science · 2020 · 2,088 citations
- Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks · SIAM Journal on Scientific Computing · 2021 · 2,044 citations
- Physics-informed neural networks (PINNs) for fluid mechanics: a review · Acta Mechanica Sinica · 2021 · 1,971 citations
- When and why PINNs fail to train: A neural tangent kernel perspective · Journal of Computational Physics · 2021 · 1,409 citations
- A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics · Computer Methods in Applied Mechanics and Engineering · 2021 · 1,290 citations
- Physics-Informed Neural Networks for Heat Transfer Problems · ASME Journal of Heat and Mass Transfer · 2021 · 1,250 citations
- Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems · Computer Methods in Applied Mechanics and Engineering · 2020 · 1,214 citations
Most recent work
- Physics-informed neural networks in water and wastewater systems: a critical review · Water Research · 2026
- Physics-informed neural networks based digital volume correlation for displacement and strain measurements · Mechanical Systems and Signal Processing · 2026
- Discovering interpretable structural dynamic response using physics-informed Kolmogorov-Arnold network · Engineering Applications of Artificial Intelligence · 2026
- Failure mechanisms and resolution in deep energy method · International Journal of Mechanical Sciences · 2026
- Koopman Operators in Robot Learning · IEEE Transactions on Robotics · 2026
- Artificial Intelligence For Partial Differential Equations In Computational Mechanics: A Review · Applied Mechanics Reviews · 2026
- Data-driven inherent strain prediction and inverse deformation control for butt welds · Welding in the World · 2026
- Multi-source heterogeneous data-driven interpretable model based on transformer and kolmogorov-arnold networks for streamflow prediction · Expert Systems with Applications · 2026
- PINN-based numerical modeling of MHD flows in porous media over linear stretching boundaries · International Communications in Heat and Mass Transfer · 2026
- A physics-informed neural network framework for state estimation of latent heat thermal energy storage: temperature reconstruction and state-of-charge prediction · Energy Conversion and Management · 2026
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