Size is Not the Solution: Deformable Convolutions for Effective Physics Aware Deep Learning

TL;DR

This work addresses the limited returns from simply increasing model size in physics-aware deep learning (PADL) and introduces deformable physics-aware recurrent convolutions (D-PARC) that embed Hybrid Lagrangian-Eulerian principles into neural networks. By learnable offsets in deformable convolutions, D-PARC achieves adaptive, task-relevant sampling—an active filtration mechanism guided by local flow dynamics—that concentrates computational focus on high-strain features while reducing effort in smooth regions. Across Burgers' equation, Navier–Stokes, and energetic-materials simulations, D-PARC attains higher fidelity than substantially larger fixed-kernel networks, with notable gains in hotspot geometry and multiscale flow features, while using far fewer parameters than the deeper baselines. These results suggest that physics-informed architectural design enabling adaptive sampling can outperform brute-force scaling, offering a practical path forward for efficient, generalizable PADL models in complex dynamical systems.

Abstract

Physics-aware deep learning (PADL) enables rapid prediction of complex physical systems, yet current convolutional neural network (CNN) architectures struggle with highly nonlinear flows. While scaling model size addresses complexity in broader AI, this approach yields diminishing returns for physics modeling. Drawing inspiration from Hybrid Lagrangian-Eulerian (HLE) numerical methods, we introduce deformable physics-aware recurrent convolutions (D-PARC) to overcome the rigidity of CNNs. Across Burgers' equation, Navier-Stokes, and reactive flows, D-PARC achieves superior fidelity compared to substantially larger architectures. Analysis reveals that kernels display anti-clustering behavior, evolving into a learned "active filtration" strategy distinct from traditional h- or p-adaptivity. Effective receptive field analysis confirms that D-PARC autonomously concentrates resources in high-strain regions while coarsening focus elsewhere, mirroring adaptive refinement in computational mechanics. This demonstrates that physically intuitive architectural design can outperform parameter scaling, establishing that strategic learning in lean networks offers a more effective path forward for PADL than indiscriminate network expansion.

Figures (17)

• Figure 1: Deformable convolution mechanics and collective scaffolding in D-PARC. Individual Kernel Analysis (a-e):a) Fixed $3 \times 3$ convolution kernel with uniform sampling at grid points. b) Deformable kernel with learned spatial offsets (blue circles) and interpolation cells (gray squares); gold element highlighted for detailed analysis. c) Bilinear interpolation for the highlighted element, showing weighted contributions from four neighboring grid cells ($Q_{11}$, $Q_{21}$, $Q_{12}$, $Q_{22}$). d) Offset vectors (blue arrows) quantifying displacement from fixed to deformed positions; formula shows magnitude calculation. e) Convex hull area (blue shading) measuring spatial extent of the deformed kernel. Multiple Kernels Analysis (f-h):f) Original positions of nine independent kernels, each color-coded for tracking. g) Deformed positions of all kernels overlaid on interpolation density heatmap, showing spatial redistribution. h) Collective interpolation density map quantifying the cumulative sampling intensity across all kernels; darker regions indicate computational resource concentration where multiple deformed kernels cluster.
• Figure 2: Deformable kernels dynamically adapt receptive fields in response to local strain dynamics. Visualization of D-PARC kernel behavior during shock-induced pore collapse in EM simulation a) Strain rate magnitude field ($\dot{\epsilon}$) at three timesteps (t = 1.36ns, 1.87ns, 2.38ns) following shock passage. Colored squares mark three spatial locations: Loc(45,97) in low strain region (magenta), Loc(153, 46) at shock front (red), and Loc (209, 67) ambient to high strain (white). b) Zoomed views ($40\times40$ pixel regions) of the three locations, revealing localized strain patterns; colored outline indicate $3\times3$ kernel with nine elements. c) Conventional $3\times3$ convolution kernel with nine elements (colored circles) at fixed grid positions. d) Deformed kernel configurations at the three locations across timesteps. Colored circles represent kernel element positions after learned spatial offsets; arrows show displacement vectors from fixed positions (panel c). Gray gridlines indicate underlying Eulerian grids. Kernels expand substantially in high-strain regions (e.g., Loc (153, 46) at t=1.87ns) while remaining compact in low-strain regions (Loc (45, 97), demonstrating adaptive context window based on local flow dynamics.
• Figure 3: D-PARC concentrates computational resources on physically salient flow features. Temporal evolution of resource allocation and physical fields during shock-induced pore collapse in energetic material simulation across four timesteps ($t = 0.34$, $1.02$, $1.70$, $2.38$ ns). a, Learned resource allocation quantified by cumulative interpolation density (number of times each pixel is sampled by deformed kernels). Diverging colormap centered at 9 (baseline for conventional $3\times3$ convolution sampling each pixel nine times); values $>9$ (red) indicate computational hotspots where D-PARC clusters multiple kernels. b, Binary microstructure map (white = solid material, black = void/pore). c, Temperature field (K) showing hotspot formation and growth. d, Pressure field (GPa) displaying shock propagation and material response. e, Strain rate magnitude (ns$^{-1}$) quantifying local deformation rates. D-PARC autonomously concentrates resources (panel a) on pore boundaries, incoming shock fronts, and emerging blast waves---regions exhibiting steep gradients in temperature, pressure, and strain rate. Resource allocation intensifies at leading edges where gradients are steepest while remaining minimal in smooth regions and domain boundaries. No prior physical guidance was provided; allocation patterns emerge purely from data-driven learning.
• Figure 4: D-PARC demonstrates regime-dependent active filtration through spatially localized effective receptive fields. Effective receptive field (ERF) visualization for energetic material simulation at $T=6$, comparing D-PARC, PARCv2, and PARCv2-L. a, Temperature field (K) with sampled locations: high-strain regions (red squares, $n=3$) and low-strain regions (white circles, $n=3$). b, Representative high-strain ERF heatmaps at gradient threshold $\tau=1\%$. Log-scale colormap shows normalized gradient magnitude; brighter regions indicate stronger influence. D-PARC exhibits compact ERF; PARCv2-L displays broad, diffuse influence. c, Representative low-strain ERF heatmaps at $\tau=1\%$. D-PARC contracts dramatically in smooth regions (492$\pm$335 pixels); PARCv2-L counterintuitively expands (4442$\pm$3322 pixels). d, Active filtration (log-log scale) across gradient thresholds (0.5--10%). D-PARC achieves lowest sparsity (highest selectivity) in both regimes. Lower curves indicate more selective resource allocation. See Table \ref{['tab:erf_threshold_sensitivity']} for complete metrics.
• Figure 5: D-PARC architecture integrates deformable convolutions with physics-aware recurrent learning. Diagram of D-PARC's hybrid Lagrangian-Eulerian approach for temporal field prediction. a, Input physical fields at $t=0$ on Eulerian grid (multiple state variables stacked as channels). b, Recurrent neural network processes fields through learned physics-aware operators. c, Feature maps determine adaptive sampling locations based on local flow dynamics, decoupled from main computation path. d, Deformable convolution kernels sample at learned offset positions across the domain. Colored circles represent individual kernel elements displaced from fixed grid positions (top row) to adaptive locations (bottom row); arrows indicate offset vectors. Each spatial location has a unique kernel configuration. e, Bilinear interpolation at offset locations followed by learned weights produces predicted fields projected back onto Eulerian grid. f, Output fields at $t=1$ maintain structured grid representation for next timestep. The architecture enables Lagrangian-type kernel transport (panel d) while maintaining computational efficiency of Eulerian representations (panels a, e, f), combining adaptive sampling with fixed-grid structure.