Traditional mesh-based CFD pipelines can become computationally demanding when mesh generation is difficult

The gap

Evidence profile

Sourced from the future work and stated research gap of the source papers, classified as general, spanning 5 journals. Those papers have been cited 15 times in total.

Supporting evidence — 5 representative gaps

• Artificial Intelligence For Partial Differential Equations In Computational Mechanics: A Review (2026) · Applied Mechanics Reviews · cited 15× · 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 path- independent and can be directly formulated as a nonlinear equation system. (2) Complex Phenomena with Inadequate Understanding: For such problems, due to their complex- ity, 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 hu- manity’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. Mo

  generalfuture work
  Keywords: nonlinear problems mechanics neural networks equation pdes computational theoretically experimental physical equations constitutive operator good
• From Newton to Neural Networks: A Review of Data-Driven Physical Modelling and the Rise of Physics-Informed AI (2026) · International Journal of Multidisciplinary Research and Analysis · doi

  10.1 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. Through su

  generalfuture work
  Keywords: scientific learning machine modelling physical approaches computational physics based simulation laws science models classical mechanics
• A Physics-Informed Neural Network Approach to Numerical Solution of Partial Differential Equations (2026) · Zenodo (CERN European Organization for Nuclear Research) · doi

  In the future, the physics-informed learning community may take on many directions beyond the vanilla PINN. In contrast, operator learning (DeepONet, FNO) moves the goalposts from finding a single solution to the solution map itself, spreading the cost of training over families of problems (Lu et al., 2021; Li et al., 2021). A hybrid solver–network approach, where classical solvers are used to correct or precondition PINNs, is expected to achieve a combination of the best of both worlds. To obtain a PINN that can adapt to variants of the PDE, with different coefficients or boundary conditions, meta-learning approaches are used. Bayesian PINNs, ensembles, and dropout are critical for safe applications ( Yang et al., 2021). Finally, there is a need for systematic benchmarking efforts, similar to ImageNet in computer vision, to allow comparing the various PINN variants that are emerging (Cuomo et al., 2022; Grossmann et al., 2024). CONCLUSION Physics-informed neural networks have evolved from an elegant idea to a wide and rapidly expanding science and machine learning discipline. PINNs avoid many of the drawbacks of classical mesh-based solvers, enable the unification of forward and inverse problems, and provide differentiable surrogates that seamlessly fit into data-driven workflows by embedding PDE residuals directly in the loss function of a NN and leveraging automatic differentiation. Applications have been extended to fluid, solid, thermal, biomedical and geophysical systems; and the framework has evolved into a family of variants, such as XPINN, cPINN, VPINN, B-PINN, DeepONet, and FNO, each tailored to overcome specific limitations of the original formulation. However, PINNs are not a one-size-fits-all solution to the classical method. There still exist some obstacles such as training cost, spectral bias, long-time integration and no complete convergence theory. The next evolution of PINNs will lie in continued advancements in adaptive sampling, causal training, hybrid solver–network approaches, and rigorous benchmarking, which will decide if PINNs are a commonplace computational science tool or a powerful yet specialized one. In either case the field of numerical PDEs will be enriched over the past decade. American Journal of Multidisciplinary Knowledge Insights • Vol. 05, No. 02 (2026) Page 20 VOLUME 05 ISSUE 02 2026 BI-ANNUAL REFERENCES Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). Automatic differentiation in machine learning: A survey. Journal of Machine Learning Research, 18(153), 1–43. Cai, S., Wang, Z., Fuest, F., Jeon, Y. J., Gray, C., & Karniadakis, G. E. (2021). Flow over an espresso cup: Inferring 3-D velocity and pressure fields from tomographic background-oriented Schlieren via physics-informed neural networks. Journal of Fluid Mechanics, 915, A102. https://doi.org/10.1017/jfm.2021.135 Chaudhari, M. (2026). Defect-sparse 2D carbon in epoxy powder coatings: Pore control and EIS durability. Genetics and Molecular Research, 25(12s). https://doi.org/10.4238/sf39b924 Chaudhari, M. (2026). Epoxy barrier coatings reinforced with solidified-waste microfillers for aggressive acids. Genetics and Molecular Research, 25(12s). https://doi.org/10.4238/96y1s945 Chen, Y., Lu, L., Karniadakis, G. E., & Dal Negro, L. (2020). Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express, 28(8), 11618–11633. https://doi.org/10.1364/OE.384875 Cuomo, S., Di Cola, V. S., Giampaolo, F., Rozza, G., Raissi, M., & Piccialli, F. (2022). Scientific machine learning through physics-informed neural networks: Where we are and what's next. Journal of Scientific Computing, 92(3), 88. https://doi.org/10.1007/s10915-022-01939-z Grossmann, T. G., Komorowska, U. J., Latz, J., & Schönlieb, C.-B. (2024). Can physics-informed neural networks beat the finite element method? IMA

  generalfuture work
  Keywords: learning physics informed pinns neural networks https pinn machine journal solution training problems classical variants
• Learning Nonlinear Finite Element Solution Operators Using Multilayer Perceptrons and Energy Minimization (2026) · SIAM Journal on Scientific Computing · doi

  We have presented a machine learning framework for learning solutions to a class of PDE problems. A core idea of the framework is to learn the corresponding discrete solution of some standard numerical method instead of aiming for the exact solution. The reason being that the standard method could be used to aid and enhance the framework. This core idea can in general be applied to various machine learning frameworks and standard methods but here we have considered a simple MLP-architecture together with energy minimization for the framework and FEM as the standard numerical method. We have presented both theoretical results (approximation error estimate) and practical applications (Newton’s method) that demonstrate how the framework may be beneficially combined with FEM. We have also presented pure framework results that show strengths and limitations of it as well as potential applications. These results are the learning errors, the usage of batches of elements for the energy during training, and the computation of quantities of interest. Concerning avenues for future work, besides looking into more advanced network ar- chitectures and training algorithms, the last elasticity example provides a natural starting point. We note that in this example (the extreme bending case), the neural network is limited by the fact that the external forces are always applied to the initial state (the unbent beam). This could be improved by introducing time dependency where the forces are allowed to change during the bending process. This would also most likely mean that the network needs to take the current position of the beam as an input parameter. This is an interesting path for future research. Acknowledgement. This research was supported in part by the Swedish Research Council Grant No. 2021-04925 and Grant No. 2022-03543, and the Swedish Research Programme Essence.

  generalfuture work
  Keywords: framework learning standard presented network machine core idea solution numerical applied energy applications training future
• Spatio-temporal uncertainty-modulated physics-informed neural networks for solving hyperbolic conservation laws with strong shocks (2026) · Engineering Analysis with Boundary Elements · doi

  Traditional mesh-based CFD pipelines can become computationally demanding when mesh generation is difficult or when the governing flow problem becomes high-dimensional. Physics-Informed Neural Networks (PINNs) often struggle to resolve shock-dominated hyperbolic conservation laws because the optimization is dominated by highly localized and highly unbalanced gradients near discontinuities.

  generalstated research gapevidence 5/5
  Keywords: traditional mesh-based cfd pipelines become computationally demanding mesh

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