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Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems
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Proposes INN-TD, a tensor-decomposition model whose univariate factors are C-HiDeNN finite-element-style interpolation functions, used for data-driven training, data-free PDE solving, and inverse optimization.
Problem
Deep learning surrogates and PINNs for large-scale, high-dimensional parametric PDEs are computationally expensive, inaccurate, and black-box, limiting use where high precision and interpretability are needed.
Approach
Combine C-HiDeNN locally supported finite-element-style interpolation bases with CP tensor decomposition so each univariate factor is a sparse interpolant; training uses boosting or all-at-once, and solving uses a Galerkin weighted-residual formulation with subspace iteration and sparse direct solvers, avoiding curse-of-dimensionality collocation.
Key results
- In the 2D Poisson solve on [0,1]^2, INN-TD reaches relative L2 error 1.754e-8 in 0.81 s with 256 parameters, versus 4.63e-3 for CP-PINN and 9.44e-5 for KAN.
- On the parametric heat-transfer data with 100% training data, INN-TD achieves test RMSE 0.129e-3 versus 0.534e-3 for MLP, 1.66e-3 for SIREN and 4.26e-3 for KAN.
- Inverse optimization of (k, P) yields relative L2 error 2.18e-3 on the target field and parameter errors around 2.7e-3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use INN-TD instead of MLP | Parametric heat PDE surrogate · Training 10% | Test RMSE (×10^-3): 3.660 -> 0.748 | Helped |
| Use INN-TD instead of MLP | Parametric heat PDE surrogate · Training 30% | Test RMSE (×10^-3): 0.912 -> 0.154 | Helped |
| Use INN-TD instead of MLP | Parametric heat PDE surrogate · Training 100% | Test RMSE (×10^-3): 0.534 -> 0.129 | Helped |
| Use INN-TD instead of KAN | Poisson Case 1 solver · [0, 1]^2 | Relative L2 norm error: $9.44 \times 10^{-5}$ -> $1.754 \times 10^{-8}$ | Helped |
| Use INN-TD instead of KAN | Poisson Case 1 solver · [0, 1]^2 | GPU wall time (sec): $56.16$ -> $0.81$ | Helped |
| Use INN-TD instead of KAN | Poisson Case 1 solver · [0, 12]^2 | Relative L2 norm error: $7.8 \times 10^{-1}$ -> $4.77 \times 10^{-4}$ | Helped |
| Use INN-TD instead of KAN | Poisson Case 1 solver · [0, 12]^2 | GPU wall time (sec): $60.68$ -> $0.22$ | Helped |
| Optimize all modes simultaneously instead of adding them through boosting (ablation) | INN-TD data-driven trainer · Training tasks | accuracy | Helped |
| Increase the number of model parameters (ablation) | INN-TD data-free solver · Poisson Case 1 | error | Helped |
Surprising findings
- In the [0,12]^2 Poisson case, KAN with 3,188 parameters reaches only 0.28 relative error, and CP-PINN errors do not decrease with more collocation points.
Limitations
- Several baseline comparisons are single-setup and the KAN training required batch-size changes that the authors note can lead to NaN losses.
- Interpretability and convergence claims are largely qualitative; error bounds are cited rather than shown here.
Experimental setup
Datasets: FEA-generated parametric heat transfer simulation data
Benchmarks: parametric heat transfer dataset (FEA) 2D/5D/10D Poisson equation with analytical solutions 2D Poisson with local source S-P-T heat transfer PDE vs implicit FEM
Metrics: RMSE relative L2 norm error GPU wall time VRAM usage
Baselines: MLP SIREN KAN CP-PINN PINN
Builds on: C-HiDeNN CP decomposition PINN proper generalized decomposition
Closest papers by meaning
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.88)
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.87)
- Towards General Neural Surrogate Solvers with Specialized Neural Accelerators (ICML 2024 · similarity 0.87)
- Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks (ICML 2025 · similarity 0.86)
- Learning from Integral Losses in Physics Informed Neural Networks (ICML 2024 · similarity 0.86)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.86)
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