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arxiv: 2606.20917 · v1 · pith:W5SFBP4Snew · submitted 2026-06-18 · 💻 cs.MS · cs.NA· math.NA· physics.comp-ph

NektarIR: A Domain-Specific Compiler for High-Order Finite Element Operations on Heterogeneous Hardware

Edward Erasmie-Jones , Giacomo Castiglioni , David Moxey This is my paper

Pith reviewed 2026-06-26 15:05 UTC · model grok-4.3

classification 💻 cs.MS cs.NAmath.NAphysics.comp-ph
keywords domain-specific compilerMLIRfinite element methodsheterogeneous hardwarespectral element methodscomputational fluid dynamicsJIT compilationhigh-order methods
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The pith

A domain-specific MLIR compiler lowers high-level finite element abstractions to optimized kernels for CPUs and GPUs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops NektarIR, a compiler using the MLIR framework, to handle high-order finite element operations on different hardware. It starts with a high-level domain abstraction that is gradually lowered to low-level code, allowing optimizations based on domain knowledge at each stage. This approach targets the common operators in spectral/hp element methods for solving partial differential equations in fluid dynamics. The goal is to reduce the effort needed for hardware-specific optimizations while maintaining performance across architectures.

Core claim

Through the NektarIR MLIR dialect and its lowering pipeline, common finite element operators are represented at a domain-specific level and compiled just-in-time to efficient code for both CPU and GPU targets, as demonstrated by comparisons with the existing Nektar++ framework.

What carries the argument

The NektarIR dialect, a custom MLIR intermediate representation for finite element operators, together with a bespoke lowering pipeline that applies domain-aware optimizations during progressive lowering.

If this is right

  • Common finite element operators can be composed into kernels for discrete differential operators.
  • These kernels can be just-in-time compiled for CPU and GPU architectures.
  • Performance is achieved without manual bespoke optimization for each hardware vendor.
  • Applications in computational fluid dynamics using spectral/hp element methods benefit from this automation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar domain-specific compilers could be developed for other scientific computing domains beyond finite elements.
  • The approach may extend to additional hardware types like accelerators beyond CPU and GPU.
  • Integration with existing frameworks like Nektar++ could be further automated for broader adoption.

Load-bearing premise

The performance of the solvers depends on a small set of common finite element operators that can be fully represented and optimized in the MLIR dialect without losing efficiency or correctness.

What would settle it

Running the generated kernels on target hardware and finding that they underperform hand-tuned implementations on the same architecture by a significant margin would disprove the claim.

Figures

Figures reproduced from arXiv: 2606.20917 by David Moxey, Edward Erasmie-Jones, Giacomo Castiglioni.

Figure 1
Figure 1. Figure 1: IR example of a matrix-matrix product kernel using the Linalg and func dialects JIT compiled elemental operator kernels via MLIR and LLVM for both CPU and GPU hardware targets. Here, we will present new MLIR passes that utilize domain-specific information to perform optimizing transformations on the IR at various abstraction levels and transform the IR from a single, high-level abstraction to hardware spec… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of the block type the final dimensions specified by the size will instead correspond to the number of quadrature points in each direction. • Finally the Layout attribute encodes the data ordering within the block, which we elaborate on below. The layout of the data is an important parameter to encode as this allows for specialised im￾plementations that can exploit specific ordering of data, partic… view at source ↗
Figure 3
Figure 3. Figure 3: Conversion from block backward transform, nir.bwd to a loop over elements and an elemental backward transform, nir.elemental_bwd. • Within the loop, a extract_slice operation returns a block corresponding to a single element in the mesh from the input to the bwd operation being transformed. • A elmnt.bwd operation is created, which returns the result of the backward transform applied to a single element. •… view at source ↗
Figure 4
Figure 4. Figure 4: Transformation from NektarIR representation of the backward transform to the affine dialect and explicit loops. The loop structure resembles the expected implementation for the backward transform as described by Equation (17). The figure contains a section of the transformed IR, with arrows indicating that the IR continues. • elmt_bwd is replaced by a collection of nested affine.for operations which repres… view at source ↗
Figure 5
Figure 5. Figure 5: The NektarIR lowering pipeline. Schematic overview of the dialects visited by the IR for an elemental operation before compilation to a particular hardware target. The dashed arrows represent the additional step required for SIMD code-generation for CPU targets. (*) The nvvm dialect is a vendor specific dialect for NVIDIA GPUs; other vendor specific dialects exist and are used to lower to their devices. su… view at source ↗
Figure 6
Figure 6. Figure 6: Loop coalescing of a triangular loop nest using loop coalescing IR transformations. The loops with non-constant upper bounds given by affine-maps are coalesced into a single loop. The upper bound of the resulting loop is obtained from an attribute attached to the innermost loop of the triangular loop nest (which is placed there when the loop is created as part of earlier transformations). Index arrays cont… view at source ↗
Figure 7
Figure 7. Figure 7: Time to lower the Helmholtz operator on hexahedral and tetrahedral elements from NektarIR to the LLVM dialect for both host and device targets. TOP and TOE refer to the two threading strategies, namely through the expansion modes and over the elements respectively. 1 2 3 4 5 6 7 8 9 Polynomial Order (p) 0.0 0.1 0.2 0.3 0.4 0.5 Time (s) Hex: Host Hex: TOP Hex: TOE Tet: Host Tet: TOP Tet: TOE [PITH_FULL_IMA… view at source ↗
Figure 8
Figure 8. Figure 8: Time to compile the Helmholtz operator on hexahedral and tetrahedral elements from NektarIR to the LLVM dialect for both host and device targets. TOP and TOE refer to the two threading strategies, namely through the expansion modes and over the elements respectively. approach and its suitability for adaptive simulations that require fast generation of new kernels as the polynomial order changes. 4.2 Runtim… view at source ↗
Figure 9
Figure 9. Figure 9: Throughput comparison of the AVX512 Helmholtz kernel in NektarIR and Nektar++ on AMD EPYC 9554 CPU. (A) and (B) show the throughput of the Helmholtz kernel on hexahedral elements while (C) and (D) correspond to the operator on tetrahedral elements. Each panel shows curves plotted on two logarithmic axes. where the number of degrees of freedom is given by the (total number of input modes)×(the number of ele… view at source ↗
Figure 10
Figure 10. Figure 10: Throughput comparison of the Helmholtz kernel in NektarIR and Nektar++ on a NVIDIA H100 GPU. (A)-(D) correspond to the threading through expansion mode method on (A)-(B) hexahedral and (C)-(D) tetrahedral elements. (E)-(H) correspond to the threading through elements method on (E)-(F) hexahedral and (G)-(H) tetrahedral elements. Each panel shows curves plotted on two logarithmic axes. ACM Trans. Math. Sof… view at source ↗
read the original abstract

Modern high performance computing (HPC) applications must target heterogeneous hardware. This requires significant work to ensure domain specific implementations translate to highly performant kernels across a range hardware types and vendors, each requiring bespoke optimization to make use of the specific target architecture. Through the development of a domain specific compiler built with the multi-level intermediate representations (MLIR) project, one can express a high-level, close to the specific domain, abstraction that is progressively lowered to a low, close to metal, abstraction. At each intermediate representation (IR), appropriate optimizations can be applied without costly analysis due to the knowledge embedded in the domain specific IRs. We apply this method to the construction of discrete differential operators for use in spectral/hp element method solvers for computational fluid dynamics (CFD). Here, the performance is driven by a small set of common finite element operators that are composed to create kernels for the discrete differential operators used to solve weak partial differential equations. We create our own MLIR dialect to represent these operators and implement a bespoke lowering pipeline to facilitate the just-in-time compilation of these kernels for both CPU and GPU architecture and illustrate performance comparisons with the Nektar++ spectral/hp element framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces NektarIR, a custom MLIR dialect for representing high-order finite element operators (mass, stiffness, etc.) in spectral/hp element methods for CFD. It describes a bespoke lowering pipeline that progressively lowers domain-specific IRs to CPU/GPU targets via MLIR, claiming that embedded domain knowledge enables optimizations without costly analysis and that the approach yields performance comparable to the hand-written kernels in the Nektar++ framework.

Significance. If the central preservation claim holds, the work would demonstrate a practical route to portable, high-performance FE kernels on heterogeneous hardware by leveraging MLIR's multi-level IR structure, potentially reducing the engineering effort required for architecture-specific tuning in spectral/hp solvers.

major comments (2)
  1. [Abstract] Abstract: the assertion that 'performance comparisons with the Nektar++ spectral/hp element framework' are illustrated is unsupported by any quantitative data, error analysis, benchmark tables, or figures; without these, the claim that the NektarIR lowering retains efficiency and correctness relative to reference operators cannot be evaluated.
  2. [Abstract] Abstract: the description of the NektarIR dialect and 'bespoke lowering pipeline' supplies neither the dialect operation definitions, the lowering rules between IR levels, nor any verification (e.g., mathematical equivalence checks or floating-point reproducibility tests) that the generated kernels for discrete differential operators remain equivalent to those in Nektar++; this directly undermines the weakest assumption that common FE operators can be captured and lowered without loss of efficiency or correctness.
minor comments (1)
  1. [Abstract] The abstract refers to 'a small set of common finite element operators' but does not enumerate them or indicate which subset is implemented in the dialect.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments correctly identify areas where the abstract overstates the manuscript's content without sufficient supporting material. We will revise the manuscript to address these issues directly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that 'performance comparisons with the Nektar++ spectral/hp element framework' are illustrated is unsupported by any quantitative data, error analysis, benchmark tables, or figures; without these, the claim that the NektarIR lowering retains efficiency and correctness relative to reference operators cannot be evaluated.

    Authors: We agree that the abstract should not assert that performance comparisons are illustrated without quantitative support being evident. The current manuscript text provided does not include benchmark tables, figures, or error analysis to back this claim. We will revise the abstract to remove or qualify the unsupported assertion and add a concise summary of performance results (including key metrics and references to tables/figures) along with basic error analysis in the revised version. revision: yes

  2. Referee: [Abstract] Abstract: the description of the NektarIR dialect and 'bespoke lowering pipeline' supplies neither the dialect operation definitions, the lowering rules between IR levels, nor any verification (e.g., mathematical equivalence checks or floating-point reproducibility tests) that the generated kernels for discrete differential operators remain equivalent to those in Nektar++; this directly undermines the weakest assumption that common FE operators can be captured and lowered without loss of efficiency or correctness.

    Authors: The referee is correct that the abstract provides no operation definitions, lowering rules, or verification evidence. While the full manuscript describes the dialect and pipeline at a high level, it does not supply the requested specifics or equivalence tests. We will revise by adding an appendix or expanded section with sample NektarIR operation definitions, key lowering rules, and verification results (e.g., mathematical equivalence and reproducibility tests) to substantiate the claims. revision: yes

Circularity Check

0 steps flagged

No circularity; implementation description relies on external MLIR framework

full rationale

The paper presents an engineering effort to build a new MLIR dialect (NektarIR) and lowering pipeline for representing and compiling finite-element operators. No mathematical derivation chain, fitted parameters, or predictions are claimed. The central assertions concern the feasibility of capturing common operators (mass, stiffness, etc.) in the dialect and lowering them without loss of correctness or efficiency; these are supported by reference to the external MLIR project rather than any self-referential construction or self-citation load-bearing step. The work is therefore self-contained as an implementation report.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the MLIR infrastructure and the assumption that finite element operators form a small composable set suitable for domain-specific IR representation. No free parameters or invented physical entities are introduced.

axioms (1)
  • domain assumption MLIR provides a suitable multi-level IR framework for embedding domain knowledge and applying staged optimizations without costly analysis.
    Invoked in the description of the compiler architecture and lowering pipeline.
invented entities (1)
  • NektarIR MLIR dialect no independent evidence
    purpose: To represent high-order finite element operators at a domain-specific level.
    New dialect created by the authors for this purpose.

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Reference graph

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