The review, led by Mohammad Nooraiepour of the University of Oslo with colleagues from the University of Bergen, Siemens Energy AI Lab, and Friedrich-Schiller-University Jena, does not declare a winner in the contest between traditional solvers and neural networks. Instead, it delivers a more unsettling and more useful conclusion: the two paradigms are epistemologically different kinds of machinery, and confusing them leads to both overhyped claims about artificial intelligence and underuse of what it genuinely offers. Classical methods are deductive, meaning their errors can be bounded by quantities derived directly from the structure of the equation and the parameters of the discretization. Machine learning methods are inductive, meaning their accuracy depends on statistical proximity to the data they were trained on. That single distinction, the authors argue, determines not which approach is superior but what each can certify, and therefore where each should be deployed.

To make the comparison rigorous, the review organizes the entire field around six fundamental computational challenges: high dimensionality, nonlinearity, geometric complexity, discontinuities, multiscale phenomena, and multiphysics coupling. These six axes function as a common testing ground. Classical methods, including finite difference, finite element, finite volume, and spectral discretizations, are assessed on each axis for their structure-preserving properties, their rigorous convergence theory, and their scalable solver design. The verdict is nuanced. On problems of moderate dimension with well-behaved geometry, classical methods remain unmatched, delivering certified accuracy with error estimates that engineers can take to a regulator or a certification authority. But the review is candid about their persistent weaknesses: the curse of dimensionality crushes grid-based methods as the number of variables grows, and geometrically complex domains can make mesh generation a bottleneck that consumes more human effort than the solve itself.

Machine learning approaches are then subjected to exactly the same critical assessment, which is itself a departure from much of the existing literature, where neural methods are typically benchmarked only against other neural methods. The review builds a taxonomy organized by the degree to which physical knowledge is incorporated into the learning process. At one end sit purely data-driven surrogates; in the middle sit physics-informed neural networks, which embed the residual of the governing PDE directly into the loss function so that the network is penalized for violating the physics; and further along sit neural operators, which learn mappings between function spaces rather than pointwise values, promising generalization across families of related problems. Graph-based architectures, transformers, and generative models round out the landscape, alongside hybrid frameworks that interleave classical solvers with learned components.

The critical findings here are sobering for anyone hoping that neural networks will simply replace finite element solvers. Physics-informed neural networks, despite enormous popularity, struggle with stiff problems, sharp gradients, and the kind of discontinuous behavior that dominates real multiphysics applications such as fracture propagation or salt precipitation during carbon dioxide storage. Their training is often fragile, their convergence theory is far less mature, and their accuracy guarantees are fundamentally statistical rather than deterministic. Neural operators fare better on generalization across parameter ranges, but the review emphasizes that their reliability is always conditional on the training distribution. A neural surrogate that performs beautifully within the range of parameters it has seen can fail catastrophically, and silently, the moment it is asked to extrapolate. Classical methods never fail silently in quite the same way, because their error bounds degrade predictably with refinement.

What elevates the review above a simple scorecard is its treatment of complementarity. The authors identify three genuine synergies between the paradigms. First, machine learning can accelerate the inner workings of classical solvers, for example by learning preconditioners, coarse-grid corrections, or adaptive mesh indicators, while the outer classical framework retains its convergence guarantees. Second, classical methods can generate the high-fidelity training data and validation benchmarks that neural surrogates require, creating a pipeline in which the deductive method certifies the inductive one. Third, learned models can capture constitutive behavior or subgrid physics that is too expensive or too poorly understood to resolve explicitly, feeding these closures into structure-preserving classical schemes. In each case, the division of labor follows the epistemological split: the learned component supplies speed and flexibility where statistics suffice, and the classical component supplies certification where proof is required.

The review goes further and develops design principles for hybrid methods, including a framework for what the authors call the structure inheritance problem, that is, how to ensure that a learned component does not destroy the conservation laws, stability properties, or symmetry structures that a classical discretization was carefully constructed to preserve. They also propose an error budget decomposition that separates the total error of a hybrid simulation into three contributions: the discretization error of the classical scheme, the approximation error of the neural component, and the coupling error introduced at their interface. This decomposition matters because it turns hybrid design from an art into an accounting exercise. If a coupled simulation goes wrong, the framework tells practitioners where to look, and it prevents the common failure mode in which a neural component’s unquantified error is silently absorbed into an apparently converged classical result.

Looking forward, the authors assess emerging frontiers with the same skeptical lens. Foundation models for scientific computing, inspired by the large language models that transformed natural language processing, are evaluated against the structural constraints of PDE solving, and the review asks whether the scaling laws that powered language models can plausibly transfer to problems governed by conservation laws and boundary conditions. Differentiable programming for inverse design, in which entire simulation pipelines are made end-to-end differentiable so that gradients can flow from design objectives back through the solver, is judged promising but dependent on the same error-accounting discipline. Quantum algorithms for PDEs are treated with particular caution: the review distinguishes between barriers that are fundamental and those that are merely contingent on engineering progress, and finds that for several quantum approaches the current obstacles look structural rather than incidental. Exascale co-design, the joint development of algorithms and hardware at extreme scale, is identified as a domain where hybrid classical-learned solvers may find their most immediate practical impact.

The timing of this synthesis is significant. Scientific machine learning has grown from a niche curiosity into a field with its own conferences, journals, and funding programs, and the hype cycle has produced both genuine breakthroughs and inflated claims. Reviews that merely catalog methods multiply yearly, but rigorous evaluations that apply identical criteria to both paradigms remain rare. By insisting that epistemological character, rather than computational speed, is the primary criterion for responsible method selection, the review provides something the community has lacked: a principled basis for deciding, problem by problem, whether a learned surrogate, a certified classical solver, or a carefully engineered hybrid is the right tool. The stakes are concrete. Applications such as geological carbon storage, where the authors’ own research is anchored through projects on salt precipitation and convective mixing during CO2 injection, demand simulations that are simultaneously fast enough for uncertainty quantification over thousands of realizations and trustworthy enough to inform regulatory decisions about subsurface storage integrity.

The conclusion the authors reach is not that one paradigm supersedes the other, but that their principled integration offers the most credible path toward computationally tractable solutions to the most consequential open problems that neither approach can address alone. For decades, computational science advanced by refining discretizations, solvers, and hardware within a single deductive tradition. The last several years added a second tradition with fundamentally different guarantees. The review’s lasting contribution may be its refusal to choose between them, and its demonstration that the future of simulation lies in knowing precisely what each tradition can promise, what it cannot, and how to combine them without letting the strengths of one conceal the weaknesses of the other. In an era when artificial intelligence is routinely described as a replacement for existing tools, this is a refreshingly precise message: the equations that govern the physical world will be solved not by machines alone, nor by classical mathematics alone, but by an alliance whose terms are now, finally, being written down.

Subject of Research: A critical synthesis of classical numerical, machine learning, and hybrid methods for solving partial differential equations

Article Title: Partial differential equations in the age of machine learning: a critical synthesis of classical, machine learning, and hybrid methods

Article References: Nooraiepour, M., Both, J. W., Kadeethum, T., & Sadeghnejad, S. (2026). Partial differential equations in the age of machine learning: a critical synthesis of classical, machine learning, and hybrid methods. Artificial Intelligence Review. https://doi.org/10.1007/s10462-026-11677-5

Image Credits: AI Generated

DOI: 10.1007/s10462-026-11677-5

Keywords: partial differential equations, machine learning, physics-informed neural networks, neural operators, finite element methods, hybrid computational methods, scientific machine learning, structure-preserving discretization, error quantification, multiphysics simulation, foundation models, carbon dioxide storage