Organizations: Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, India · Department of Computer Science & Engineering, IIT Kharagpur, Kharagpur, India
Abstract
Spreading of liquid droplets on solid substrates constitutes a classic multiphysics problem with widespread applications ranging from inkjet printing, spray cooling, to biomedical microfluidic systems. Yet, accurate computational fluid dynamic (CFD) simulations are prohibitively expensive, taking more than 18 to 24 hours for each transient computation. In this paper, Physics-Informed Laplace Operator Neural Network (PI-LNO) is introduced, representing a novel architecture where the Laplace integral transform function serves as a learned physics-informed functional basis. Extensive comparative benchmark studies were performed against five other state-of-the-art approaches: UNet, UNet with attention modules (UNet-AM), DeepONet, Physics-Informed UNet (PI-UNet), and Laplace Neural Operator (LNO). Through complex Laplace transforms, PI-LNO natively models the exponential transient dynamics of the spreading process. A TensorFlow-based PI-LNO is trained on multi-surface CFD data spanning contact angles θsε[20,160], employing a physics-regularized composite loss combining data fidelity (MSE, MAE, RMSE) with Navier-Stokes, Cahn-Hilliard, and causality constraints.
This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned 1×1 convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight αwav, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.
Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs). However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are (i) distribution-free with provable coverage guarantees, and (ii) spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable (82-100%). FNO outperforms CNN and DeepONet by 10-12x.
Dental aerosol procedures produce sub-50 micrometre nuclei that can remain airborne for long periods in enclosed clinics, creating pathways for airborne pathogen transmission. Reynolds-Averaged Navier-Stokes (RANS) simulations with Euler-Lagrange particle tracking capture this transport accurately but require very long run times per scenario, which precludes real-time clinical decision support in 3D. We present the Eulerian-Lagrangian Graph Interaction Network (ELGIN), a physics-informed graph surrogate that jointly predicts carrier-flow dynamics on the OpenFOAM polyhedral mesh and the per-parcel motion of the polydisperse spray cloud. ELGIN couples a multi-head Graph Transformer with Jacobi-preconditioned learnable pressure projection and a turbulence-closure head to a sigmoid-gated Lagrangian Interaction Network through differentiable inverse-distance mesh-parcel coupling, and advances parcels with a symplectic Stormer-Verlet integrator. A four-stage physics-informed curriculum stabilises 260-step autoregressive rollouts without gradient explosion. A parameter sweep with foam-extend 4.1 OpenFOAM reactingParcelFoam across clinically relevant ventilation rates and handpiece spray speeds provides CFD ground truth. This article reports a single-case demonstration in which both ELGIN and a Lagrangian-only baseline (M0) are trained and evaluated on Sweep_Case_03 of a twenty-case sweep; full 16/2/2 retraining is in progress and will replace all reported metrics. On this case, ELGIN tracks the foam-extend particle cloud much more closely than M0: mean parcel displacement error falls from 19.56% to 16.20% of room width and cloud radius-of-gyration error from 9.85% to 6.58%. A 26-second rollout completes in ~64 s on a 4 GB GPU, approximately 37x faster than the foam-extend reference pipeline, toward per-appointment infection-risk screening once the multi-case checkpoint is in place.