Discrete Solution Operator Learning for Geometry-Dependent PDEs

TL;DR

Geometry-driven changes in domain topology and boundaries create discrete, non-smooth variations that challenge smooth-function-space neural operators. DiSOL addresses this by learning a discrete solution procedure on an embedded grid, decomposing the solver into a local-contribution operator, a multiscale assembly, and an implicit solver, while applying geometry-conditioned modulation and domain projection. Across Poisson, advection–diffusion, linear elasticity, and time-dependent heat conduction, DiSOL achieves stable, accurate predictions under both ID and strongly OOD geometries, outperforming continuous baselines such as DeepONet and FNO whose performance degrades with discrete geometric changes. This work highlights a complementary, procedure-oriented bias for operator learning in geometry-dominated regimes and points to future extensions involving variable-resolution discretizations and hybrid discrete-continuous frameworks.

Abstract

Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological changes, abrupt changes in boundary conditions or boundary types, and changes in the effective computational domain, which break the smooth-variation premise. Here we introduce Discrete Solution Operator Learning (DiSOL), a complementary paradigm that learns discrete solution procedures rather than continuous function-space operators. DiSOL factorizes the solver into learnable stages that mirror classical discretizations: local contribution encoding, multiscale assembly, and implicit solution reconstruction on an embedded grid, thereby preserving procedure-level consistency while adapting to geometry-dependent discrete structures. Across geometry-dependent Poisson, advection-diffusion, linear elasticity, as well as spatiotemporal heat-conduction problems, DiSOL produces stable and accurate predictions under both in-distribution and strongly out-of-distribution geometries, including discontinuous boundaries and topological changes. These results highlight the need for procedural operator representations in geometry-dominated regimes and position discrete solution operator learning as a distinct, complementary direction in scientific machine learning.

Figures (5)

• Figure 1: Discrete Solution Operator Learning (DiSOL) in contrast to continuous neural operator paradigms.a, Classical discrete numerical methods for partial differential equations (PDEs), exemplified by the finite element method. A PDE operator can be realized through a sequence of fixed discrete procedures, including local operator evaluation (e.g., element-level stiffness and force computation), global assembly, and solution via assembled global systems. Under geometry variation, the assembled results change, whereas the underlying assembly procedures remain unchanged. b, An abstracted view of discrete problem solving, highlighting the canonical workflow shared by many numerical methods: local contributions evaluation, global information assembly, and solution through globally aggregated representations. c, Discrete Solution Operator Learning (DiSOL). Instead of explicitly constructing numerical matrices, DiSOL represents the discrete solution process through learnable neural operators that conceptually correspond to the local operator, the multiscale assembly operator, and the implicit problem-solving operator. These components collectively form a discrete operator that preserves procedure-level invariance under geometry variation, in the sense that the underlying solution procedure remains unchanged, while adapting its outputs to geometry-dependent discrete structures. d, Representative continuous neural operator paradigms, illustrated by DeepONet DeepONetLu2021 and FNO FNOLi2020. These approaches model PDE operators as global mappings in continuous function spaces, using global basis combinations or spectral transformations. Such representations are primarily designed for smooth function-space mappings, and may become misaligned when geometry-induced domain changes and discontinuities dominate the problem setting. In contrast, DiSOL does not aim to approximate continuous operators directly, but learns discrete solution procedures, leading to a fundamentally different inductive bias in geometry-dependent problem settings.
• Figure 2: Discrete solution operator learning (DiSOL) for a geometry-dependent 2D Poisson problem.a, Problem formulation and data representation. The input to the model consists of three discrete fields defined on a fixed Cartesian grid: a geometry mask, a boundary-condition selection map, and a source term. The model outputs the corresponding discrete solution pattern. b, Training and validation loss histories for DiSOL, DeepONet and the FNO under comparable model capacity and training settings. DiSOL converges faster and achieves substantially lower validation loss (All models are trained under the same optimizer/schedule and have comparable parameter counts, $\approx0.13$M each; details in Supplementary Information C.4 and 5.). c, ID test results for two representative cases. DiSOL accurately predicts the ground-truth solution patterns, while DeepONet shows pronounced distortions in the normalized pattern values (all fields are normalized patterns), despite the overall smoothness of the solutions. d, OOD generalization tests involving unseen geometries with sharp corners and internal holes, discontinuous and localized boundary conditions, and high-frequency source terms. DiSOL remains stable and consistent with the reference solutions, whereas DeepONet and FNO show substantial degradation in solution-pattern fidelity. See Supplementary Information A.1, B and E.1 for full setup and extended statistics.
• Figure 3: Discrete solution operator learning (DiSOL) results for the advection-diffusion equation.a,b, Representative ID test cases under two transport cases: a diffusion-dominated case ($\text{Pe} \approx 0.45$) and an advection-dominated case ($\text{Pe} \approx 4.5$). Ground-truth solutions are compared with predictions from the proposed discrete solution operator (DiSOL) and the FNO. As transport effects become stronger and solution structures align more closely with flow direction and complex geometric boundaries, DiSOL consistently preserves the global transport patterns and boundary-induced features, whereas FNO exhibits increasingly pronounced structural deviations, particularly near irregular geometric features. c,d, Training and validation loss histories for DiSOL, FNO, and DeepONet under the same two Péclet number settings ($\text{Pe} \approx 0.45$ and $\text{Pe} \approx 4.5$). In both cases, DiSOL converges rapidly and achieves substantially lower training and validation errors, while the continuous neural operator baselines saturate at significantly higher error levels. e, OOD generalization results for the advection-diffusion equation at $\text{Pe} \approx 4.5$, involving unseen geometries with sharp corners and topological changes, together with localized and discontinuous boundary conditions. DiSOL preserves the overall transport structure and solution morphology. f, Statistical comparison of relative L1 errors across different models under ID and OOD settings for both $\text{Pe} \approx 0.45$ and $\text{Pe} \approx 4.5$. Box plots (logarithmic scale) show that DiSOL consistently outperforms continuous neural operator baselines in both cases, and that performance degradation is primarily driven by geometric distribution shifts rather than by increased transport dominance. See Supplementary Information A.2, B and E.2 for full setup and extended statistics.
• Figure 4: Discrete solution operator learning (DiSOL) results for geometry-dependent 2D linear elasticity.a, Problem formulation and data representation for the linear elastic solid mechanics problem. The input to the model consists of five discrete channels defined on a fixed Cartesian grid, encoding the geometry mask, displacement boundary conditions in the horizontal and vertical directions, and externally applied force components. The model outputs the corresponding two-component displacement field $(\hat{u}_x,\hat{u}_y)$ defined on the same grid. b, Representative ID test results. Ground-truth displacement fields are compared with predictions from the proposed discrete solution operator (DiSOL) for two representative geometries and loading configurations. DiSOL accurately reconstructs both horizontal and vertical displacement components, capturing the global deformation patterns and boundary-induced responses without introducing spurious oscillations or rigid-body artifacts. c, OOD generalization results involving unseen geometries with increased complexity, including internal holes, topological changes, and altered boundary condition configurations. Despite the substantial changes in geometric structure and boundary constraints, DiSOL preserves the overall deformation modes and displacement distributions, maintaining consistency with the reference solutions for both displacement components. See Supplementary Information A.3, B and E.3 for full setup and extended statistics.
• Figure 5: Discrete solution operator learning (DiSOL) for the heat conduction problem.a, Problem formulation and data representation. The model input consists of discrete channels defined on a fixed Cartesian grid, encoding the geometry mask, boundary condition locations, source term distribution, and initial condition (IC). The time coordinate is provided as an additional input channel. The model predicts the temperature field at queried times $t$; during training, $t$ is sampled from $[0,20]\,\mathrm{s}$. b, Representative ID test results at $t=2\,\mathrm{s}$, $10\,\mathrm{s}$, and $20\,\mathrm{s}$, showing close agreement between ground-truth solutions and DiSOL predictions. c, Zero-shot OOD evaluation combining geometric extrapolation and temporal forecasting. The model is tested on an unseen geometry with internal voids and altered boundary configurations. Predictions at $t=30\,\mathrm{s}$, $40\,\mathrm{s}$, and $50\,\mathrm{s}$ extend beyond the training temporal interval, corresponding to genuine future forecasting scenarios. Color bars indicate normalized temperature magnitude. See Supplementary Information A.4, B and E.4 for full setup and extended statistics.