This article is a systematical review on the current numerical methods used in the numerical solution of partial differential equations (PDEs) in which our special attention is focused on the latest developments essentially around the past decade. It starts to summarize standard classifications of PDE, and briefly returns to the classical classification of discretization mixed methods, mainly finite difference, finite element, or finite volume, which can get a preliminary knowledge in order to make a scientific comparison. The review then gives an overview of more recent developments of spectral and spectral element methods, meshfree and material point methods, adaptive and multiscale formulations, and multigrid-based acceleration methods. One of the major contributions of the present article is the learning-based and probabilistic paradigm. In particular, the manuscript talks about AI-driven approaches such as physics informed neural networks that help the operator learn using DeepONet, Fourier neural operators and other hybrid approaches that combine data-driven factors and classical numerical solvers for operator learning. Across these methodological families, the article makes a qualitative comparison in terms of aspects of accuracy, computational cost, stability, scalability, implementation complexity, and representative application domains. Illustrative examples cover the fields of fluid dynamics, solid mechanics, biomedical flow modelling, as well as problems arising in the problems of finance and quantum systems.The review concludes with an open questionnaire, in particular the need for more drastic theoretical guarantees and findings on uncertainty quantification combined with the need for efficient coupling with high-performance computing environments. It describes promising avenues for future research, in particular in rigorously analyzed hybrid approaches and solvers based on operator learning.
1. Introduction
Partial differential equations (PDEs) form the very fundamental building blocks of mathematical modeling in a very wide range of domains of science and engineering, such as fluid dynamics, heat and mass transfer, elasticity, electromagnetics, as well as quantitative finance, where they describe the spatiotemporal evolution of physical fields
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During the last decades, classical discretization techniques, among others, finite difference, finite element and finite volume techniques, have enabled reliable large-scale computations in a number of application domains. However, these well-established methodologies are challenging to employ in solving complex problems that have complex geometries, highly nonlinear dynamics, and large parametric dimensionality, in which issues related to stability, the level of accuracy and calculation cost become more and more pronounced
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All these limitations have allowed the current interest in modern numerical methods that aim to improve the transfer of robustness and predictive quality while reducing the computational effort, especially through better scalability on parallel architectures and the introduction of data-driven components. Against this background, the current review is aimed to systematically review recent developments in solvers of partial differential equations reported in the period 2015 to 2025, by focusing particular attention on spectral and meshfree formulations, adaptive and multiscale approaches, artificial intelligence-based and probabilistic solvers, as well as advancing hybrid approaches, where learning modules are combined with standard numerical schemes. The manuscript outlines at first the required mathematical and numerical prerequisites before giving a brief historical overview of the main ideas of the major contemporary methodological families. This is followed by an evaluative appraisal of their respective strengths and trade-offs in terms of the key performance criteria. Representative applications are highlighted throughout the discourse, and outstanding challenges and possible avenues for future research are stated in the conclusion of the review.
2. Classical background
Simple classification and intrinsic properties of problems of partial differential equations. Typically, Partial Differential Equations fall into one of three groups — elliptic, parabolic, and hyperbolic — depending on the algebraic structure of the highest order term of the differential operator, and it is directly related to the qualitative nature of its solutions
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In the problems of electrostatics, elastic theory and potential flow dynamics, elliptic equations like the Laplace and Poisson equations are used widely. Most often these solutions are smooth, meaning that the boundary conditions affect the solution in and around the region. For example, parabolic equations, such as the heat equation, and the Fokker-Planck equation are of interest, since they model diffusive, dissipative or relaxation phenomena; in such systems, the solution evolves with time, and, by necessity, "smoothes" the irregularities, and approaches an equilibrium state, while, at the same time, remaining constrained by the set of initial data and boundary conditions. This is the control of finite-slope propagation of signals and characteristics defined by hyperbolic equations like the wave and advection equations. Their solutions may show wave-like behavior and even exhibit steep gradients and propagating fronts, or discontinuities, particularly for nonlinear regimes or more complicated forcing. From the modelling point of view, a problem of the above type is said to be well — posed if it has the properties: existence, uniqueness, and dependence on the initial and/or boundary data that is continuous with respect to those data. Well, poshness is essential for mathematical soundness as well as for numerical reliability: discretization schemes, which allegedly fail to meet these criteria, e.g., due to incompatible treatment of the boundary or inadequate stability control, can provide relevant unstable, divergent or physically implausible solutions
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For a representative
The weak formulation can be expressed as:
The derivatives naturally induce moving from the unknown to the test function, the discretization's of the above partial differential equations in Galerkin form on which the finite element family of methods
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Basic schemes for number crunching Foundational numerical methods to partial differential equations. Foundational method for numerical solution of partial differential equations (PDEs): systematic discretization of the space-time domains for computing massive algebraic systems that approximate the continuous problems. The finite difference method (FDM) involves computing difference quotients of spatial derivatives at points in a structured grid in simple stencil operations and easy to implement on regular domains
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In contrast, the FEM relies on a variational or weak formulation, and represents the solution by a patchwork of piecewise polynomial basis functions defined on a system of elements which need not be arranged in any special way and can be quite unstructured. First, such methodology issues as convergence, stability and the intrinsic complexity of the problem can be dealt with in a strict manner; second, complex geometries and different materials can be accommodated; and third, a vast array of boundary conditions can be imposed in a consistent and natural manner. In the context of finite element theory, discontinuous Galerkin (DG) methods introduce discontinuous basis functions at interfaces between elements and use numerical fluxes on interfaces to couple the elements together
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The finite volume method, on the other hand, is conservation-based, where the governing equations are put into the control volumes, and the conservation is applied through a balance of fluxes at the faces of the cells
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The classic finite difference, finite element (including discontinuous Galerkin) and finite volume methods are collectively referred to as finite difference type schemes, and new schemes using spectral, mesh-free, adaptive and AI-assisted methods are evaluated against them, particularly for accuracy, robustness and computational efficiency.
3. Modern numerical methods
Spectral and spectral element methods approximate solutions of partial differential equations by expanding the unknown field as an expansion in globally supported basis functions, usually trigonometric polynomials (e.g. Fourier bases) and families of orthogonal polynomials, and solving for the corresponding expansion coefficients from the governing equations and accompanying constraints [2]. Spectral methods approximate the solution [LATEX_FORMULA]u(x)[/LATEX_FORMULA] as a series expansion of smooth and globally supported basis functions [LATEX_FORMULA]\phi_{k}(x)[/LATEX_FORMULA]:
where [LATEX_FORMULA]N[/LATEX_FORMULA] is the number of degrees of freedom, and [LATEX_FORMULA]u^{k} \phi_{k}(x)[/LATEX_FORMULA]
The spectral collocation (or pseudo-spectral) method is the most widely used approach. The PDE is enforced exactly at a set of collocation points, such as Chebyshev Gauss Lobato points.
If the PDE is given by [LATEX_FORMULA]L u=f[/LATEX_FORMULA] where [LATEX_FORMULA]L[/LATEX_FORMULA]
Where [LATEX_FORMULA]x_j[/LATEX_FORMULA] are the collocation points. The derivatives of [LATEX_FORMULA]u_N[/LATEX_FORMULA] are computed using spectral differentiation matrices, which convert the differential operator [LATEX_FORMULA]L[/LATEX_FORMULA] into a matrix [LATEX_FORMULA]D[/LATEX_FORMULA].
Spectral schemes in their classical global form can converge very rapidly, with a high degree of smoothness of the solution they represent; many spectral schemes can be exponentially convergent with the number of modes retained. This is due to the fact that typically a small number of modes can represent the large-scale features of the solution with great fidelity
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By merging the spectral accuracy and element-wise domain decomposition, the spectral-elements approach overcomes some of the above drawbacks. The computational domain is subdivided into elements, and the solution is approximated within each element by high order polynomial functions defined on a reference element which gets mapped to the physical element
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In representing the solution field, meshfree methods do not involve any fixed connectivity mesh, but rather a set of randomly placed nodes or particles, from which the solution field is reconstructed by using a set of suitable shape functions or kernel-based interpolants
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For instance, radial basis function (RBF) methods approximate the solution using a superposition of radially symmetric basis-functions that are centered at the nodes and that can be used to achieve high-order accuracy, even for irregular node-distributions, and are relatively easy to be extended to higher dimensions
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In RBF methods, the solution [LATEX_FORMULA]u (x)[/LATEX_FORMULA] is approximated as a linear combination of radial basis functions [LATEX_FORMULA]\Phi[/LATEX_FORMULA], which depend only on the radial distance [LATEX_FORMULA]r=\left\|x-x_{i}\right\|[/LATEX_FORMULA] between the evaluation point x and the node [LATEX_FORMULA]x_i[/LATEX_FORMULA]:
Where [LATEX_FORMULA]\lambda_{i}[/LATEX_FORMULA] are the unknown coefficients, and [LATEX_FORMULA]P(X)[/LATEX_FORMULA]
Smoothed particle hydrodynamics (SPH) adopts a fully Lagrangian description in which the continuum is represented by moving particles that carry mass, momentum, and other state variables. Spatial derivatives are approximated through smoothing kernels, making SPH well-suited to large deformations, free-surface flows, and fragmentation phenomena that are challenging for mesh-based Eulerian discretization’s
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Adaptive, MultiGrid and Multiscale Methods. Adaptive approaches aim to allocate the available computational effort in those regions, which can provide the most information from existing or required detailed spatial or temporal discretization in regions of high or estimated error. This kind of principle can be realized through the technique of adaptive mesh refinement (AMR), refining and coarsening the computational mesh according to the indications of errors or error estimators that can improve the efficiency of computing the localized features like shocks, boundary layers and sharp interface without any uniform refinement of the entire region
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Multiscale methods and reduced-order models (ROMs) tackle problems, where phenomena on vastly different scales interact, or where repeated solutions of similar PDE's are required, for instance, in optimization and control problems. Variational multiscale formulations break up the solution into resolved and unresolved parts, which explicitly models the effects of the small-scale dynamics on the large-scale behavior so as to provide a systematic framework for the modelling of turbulence and subgrid scale processes. ROM techniques, e.g., based on proper orthogonal decomposition and on projection methods, give rise to low-dimensional dynamical systems that can be used to approximate the original PDE model with significantly reduced computational cost, but at the price of an offline training or basis construction phase.
Probabilistic and AI-based methods. One of the most recent developments is the use of machine learning combined with numerical analysis to create a new family of PDE solvers based on data-driven models subject to physical constraints
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Total Loss Function:
where [LATEX_FORMULA]\lambda[/LATEX_FORMULA]
PDE Residual Loss [LATEX_FORMULA]\left(L_{f}\right)[/LATEX_FORMULA]: Computed based on the PDE residual [LATEX_FORMULA]f[/LATEX_FORMULA] evaluated at random collocation points within the domain [LATEX_FORMULA]\Omega[/LATEX_FORMULA].
Derivatives are obtained via automatic differentiation.
Boundary and Initial Condition Losses [LATEX_FORMULA]\left(L_{b c} \text { and } L_{t c}\right)[/LATEX_FORMULA]
The network can automatically differentiate with spatial and temporal variables
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Hybrid strategies try to combine complementary paradigms in a manner where each paradigm's strengths complement each other and ameliorate the drawbacks of each other. One approach frequently used is to discretize the various parts of the domain differently, for instance, using spectral or high-order finite element methods in smooth regions and finite volume or discontinuous Galerkin (DG) methods in regions with discontinuities or steep gradients or in regions with very complicated geometry, where robustness and conservation are more important
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All in all, hybrid and coupled methods are a promising route towards developing effective, efficient and flexible PDE solution strategies that can make use of the well-established techniques of classical numerical analysis and the growing power of current methods based on machine learning.
4. Comparative analysis and applications
Qualitative comparison. As far as accuracy is concerned, the spectral and spectral element methods converging with the increase of the resolution, are usually the fastest, especially if the solution is sufficiently smooth, and almost exponential convergence is obtained
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Although methods such as meshfree methods and today's trading adaptive methods can be quite good at dealing with moving fronts, evolving interfaces, and sharp layers, their global error behavior, however, can be much more problem dependent and is less well known to characterize and study rigorously, and the stability and approximation properties can be sensitive to the distribution of nodes, adaptivity criteria, and choices of stabilization
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Meanwhile, on many benchmarks, the accuracy of AI-based solvers can be the same or comparable to that of their classical counterparts; and on the academic side, it is well to note that the error capacity of these solvers is often not the only source of error, but also the data-covering and the dynamics of the operators trying to get to production, and the inductive biases of the architecture.
The trade-offs are no less nuanced, when it comes to the cost of computation. Even with well-matured linear and nonlinear solvers, e.g., multigrid and/or Krylov-subspace
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In contrast, meshfree methods and discontinuous Galerkin approaches have higher assembly and memory costs because of the higher complexity of the stencils, larger number of local degrees of freedom and the price of the construction of the shape functions or flux couplings
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Spectral methods can be very competitive on simple geometries, where the rapidity of the transforms (using structured operators which do not incur a large overhead) can be very high, but when confronted with complex geometries of complicated boundaries, strong localisation and/or coefficients, this efficiency may drop drastically. In such an environment the advantages of the global representations are reduced and a higher degree of algorithmic complexity is needed
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Learning-based approaches have a specific cost model: They are likely to have high initial training costs on customized hardware but are cheap to evaluate once they have been trained. This "train one, evaluate many" paradigm is not only pivotal, but also involves a completely new way of thinking, for example, whether the training costs have to be taken into account, data and bias in the data, or mismatch between the different deployment domains if the methods are compared to classical solvers in a fair analytical framework
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In terms of stability and robustness, well-developed numerical schemes are numerically efficient due to the existence of a sound theoretical background, algorithmic constraints (e.g. CFL-conditions) and limiting and stabilization methods of nonsmoothed solutions and systematic error estimation schemes. The reliability of these methods hinges on these attributes and they should have predictable and controllable failure modes in challenging engineering applications
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In contrast to this, there are not often extensive stability theories of meshfree techniques and the corresponding solvers based on artificial neural networks. They might have non-physical oscillations, which is a problem even in mechanics, they might lose their conservation properties or have some pathologies induced by the optimization which are not easy to pre-assess in the first place: thus, a long-lasting gap between empirical performance and serious guarantees
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Yet axes of scalability and implementation complexities still make a difference in practice when evaluating such approaches. The use of local formulations such as DG, adaptive mesh refinement (AMR) pipelines, and many meshfree particle methods is inherently conducive to parallelization, due to the fact that only localized evaluations of operators and limited communication patterns are used
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To make the foregoing comparison more complete, a summary of the different regimes, geometric treatment, merits and drawbacks of representative reduced-order, meshfree, operator-learning and hybrid PDE solution paradigms is given in Table 1.
Comparative synthesis of representative PDE solution methods
| Paradigm | Core idea | Best-fit regime | Geometry / mesh handling | Key strengths | Main limitations and compute notes |
| [15] | Projection-based reduced-order model built on a high-fidelity discretization. | Repeated queries for parametrized PDEs, optimal control, and inverse problems. | Medium; strongest when parametrization and the reference model are well defined. | Certified error estimates; offline/online split; strong classical numerical grounding. | Intrusive and less flexible for rapidly changing geometries; best when the solution manifold is smooth. |
| [17] | Train a neural ansatz on randomly sampled space-time points instead of a mesh. | High-dimensional PDEs where classical grids are infeasible, especially finance and control. | Meshfree, but typically demonstrated on simpler domains rather than irregular engineering geometries. | Avoids explicit meshes; demonstrated at very high dimension in representative studies. | Per-problem training cost is high; optimization replaces, rather than removes, heavy computation. |
| [22] | Neural solution representation constrained by PDE residuals, initial conditions, and boundary conditions. | Forward and inverse problems with sparse data or strong physics constraints. | Meshfree and flexible; can encode Dirichlet, Neumann, Robin, and periodic conditions. | Data-efficient and physically informed; natural for inverse and hybrid data-physics workflows. | Training pathologies include spectral bias, gradient imbalance, and causality issues; weak on high-frequency or multiscale solutions without advanced strategies. |
| [24] | Operator learning with branch and trunk networks mapping input functions to output functions. | Many-query parametric PDE families where amortized inference matters. | Medium-high; can pair with CNN or GNN branches and is not tied to one geometry class. | Fast reuse after training; robust in noisy and complex settings. | Needs supervised training data and architecture tuning; upfront training cost can be substantial. |
| [28] | Neural operator with an integral kernel parameterized in Fourier space. | Structured-grid parametric PDE families needing fast inference. | Low-medium; strongest on regular domains and grid-like data. | Very fast inference and zero-shot super-resolution in benchmark settings. | Performance often degrades with noisy inputs or complex geometries; less natural for irregular domains. |
| [30] | Hybrid neural operator combining data supervision with PDE constraints, often at higher resolution. | Operator learning when physical validity and high-resolution fidelity matter. | Medium; inherits operator-learning structure while adding physics constraints. | Better generalization and physical validity than pure FNO, while retaining large speedups. | Still depends on operator-learning infrastructure and fine-tuning; not as geometry-native as graph-mesh approaches. |
| [33] | Graph-network simulator operating directly on meshes and allowing adaptive remeshing. | Irregular geometries, adaptive meshes, mechanics, cloth, and fluid problems. | Very high; built for unstructured irregular meshes and adaptive discretization. | Resolution-independent dynamics; adaptivity concentrates compute where gradients are strong. | Requires graph-based training data and rollout design; lacks classical certification. |
| [35] | Autoregressive message-passing solver with stability-oriented training. | Generalization across resolution, topology, geometry, discretization regularity, and boundary conditions. | Very high; graph representation supports irregular sampling, geometry, and topology changes. | Fast, stable, and accurate in 1D and 2D tests; explicitly connected to FDM, FVM, and WENO ideas. | Training remains nontrivial and error bounds are not certified; best viewed as a neural-numerical hybrid. |
| [37] | Pretrained graph-transformer plus implicit neural representation for cross-family PDE solving. | Zero-shot or few-shot transfer and inverse coefficient recovery across one-dimensional PDE families. | Currently low in geometry scope because published evidence is still largely one-dimensional. | Promising foundation-model direction with zero-shot transfer and rapid fine-tuning. | Current evidence remains mostly one-dimensional, and pretraining is heavy relative to narrower expert models. |
The solvers of the Neural PDE are supposed to be compared by the accuracy but also in regard to the hardware needs
[32][40][43][39][44][47][45]
5. Representative application areas
Computational fluid dynamics, turbulence. The high-fidelity simulation of incompressible and compressible flows is a great source of innovation for new methods of numerically solving partial differential equations. Naturally, for a large number of applications in engineering and geophysics, finite-volume and finite-element discretization are by far the most popular workhorses, largely because of the good compromise of robustness, conservation properties and geometric flexibility offered by these discretization. Their performance is often improved through the use of adaptive mesh refinement (AMR), to focus the high-resolution in dynamically significant areas, and other computational iteratively challenging reconstruction schemes, such as multigrid acceleration, to decrease the computation time for their solvers, and turbulence-modelling schemes developed in variational multiscale frameworks, which attempt to formally divide and always consistently represent the flow with the resolved and the unresolved structures
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Solid Mechanics/Fracture. Most of the simulations used in industry that involve elasticity, plasticity and wave propagation in solids are based on the classical finite element method (FEM) including the high-order and discontinuous Galerkin (DG) methods. Meshfree and material point method (MPM) formulations are very useful in problems of large deformations, impact and fracture where remeshing would be prohibitive
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Fluid mechanics of Biomedical flows and transportation. Patient-specific vascular networks, or even the transport of pharmaceutical or signaling molecules, are typically modeled via finite-volume or finite-element solutions on a complex geometry obtained from medical imaging data
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Financial Partial Differential equations and Quantum problems. In the field of quantitative finance, partial differential equations formulations play a fundamental role, and in particular, the high dimensional Black Scholes-type models for multiple asset derivatives and similar stochastic partial differential equations models of risk factors. Numerically, these problems are challenging in that the computational cost rapidly grows as the number of dimensions is increased, an effect known as the "curse of dimensionality," and the simple possibility of mesh refinement for the solution of multisets problems is limited
[1][4][5][6][44][45][46]
Uncertainty quantification (UQ) is also a current bottleneck, and while probabilistic formulations can offer valuable information with regards to credibility, risk, etc, UQ on complex PDE models is often expensive and there is a strong incentive in the research community to work towards approaches that enable realisation of such models which involve integration of probabilistic modelling and efficient sampling, reduced order modelling, operator learning surrogates, etc, in a way that will control cost without compromising the statistical reliability of the results.
6. Conclusion
Modern numerical methods for the solution of partial differential equations are a vast and evolving arsenal, and there is a large body of theoretical and experimental evidence that supports the lack of a uniform paradigm that is universally found to be best for all classes of PDEs, geometries, and definition criteria. In this regard, classical finite-difference, finite-element, and finite-volume methods remain predictors of last resort for lack of maturity, efficiency, and well-justified analytical background
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Simultaneously, spectral, meshfree, adaptive and multiscale methods greatly extend the range of practicality of PDE simulation (e.g. in attaining superior accuracy for smooth fields, higher flexibility with respect to geometrical domains and efficient resolution of localised or multiscale phenomena which cannot generally be captured by traditional uniform discretisation
[2][3][7][9])
More recently, solvers based on the Al — the most effective for large dimensions inverse and parametric problems — and operator learning methods have become influential developments, especially for high dimensional, inverse and parametric problems, for which one needs repetitive evaluation, or fast exploration of families of solutions. These methods can provide significant computational benefits and new modelling capabilities but usually do not yet provide the robustness, interpretability and theoretical guarantees supporting the routine use of mature discretization schemes in a safety-critical context
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