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Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method

Yuhe Wang, Min Wang

TL;DR

This work reframes physics-informed PDE solving as a projection-based procedure by introducing the Neural Basis Method (NBM), which builds a physics-conforming neural basis via a fixed-parameter dual-layer network and enforces PDEs through a weighted least-squares residual, yielding a stable, interpretable certificate of error. The approach is demonstrated on advective multiscale Darcian dynamics, combining a mixed Darcy–flow formulation with energy-consistent weighting and upwind-stabilized transport, achieving spectral-like accuracy and substantial speedups in parametric settings (NBM-OL). A key innovation is the physics-conforming neural vector basis, built from a Helmholtz-type decomposition to preserve mass conservation and improve stability, with extensions to 3D. The framework further provides an operator-learning pathway (NBM-OL) to map parametric inputs—such as permeability fields and boundary fluxes—to reduced coordinates, enabling robust in-distribution generalization, strong out-of-distribution robustness, and orders-of-magnitude speedups over conventional finite-volume solvers for many-query tasks in subsurface flow and transport.

Abstract

Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.

Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method

TL;DR

This work reframes physics-informed PDE solving as a projection-based procedure by introducing the Neural Basis Method (NBM), which builds a physics-conforming neural basis via a fixed-parameter dual-layer network and enforces PDEs through a weighted least-squares residual, yielding a stable, interpretable certificate of error. The approach is demonstrated on advective multiscale Darcian dynamics, combining a mixed Darcy–flow formulation with energy-consistent weighting and upwind-stabilized transport, achieving spectral-like accuracy and substantial speedups in parametric settings (NBM-OL). A key innovation is the physics-conforming neural vector basis, built from a Helmholtz-type decomposition to preserve mass conservation and improve stability, with extensions to 3D. The framework further provides an operator-learning pathway (NBM-OL) to map parametric inputs—such as permeability fields and boundary fluxes—to reduced coordinates, enabling robust in-distribution generalization, strong out-of-distribution robustness, and orders-of-magnitude speedups over conventional finite-volume solvers for many-query tasks in subsurface flow and transport.

Abstract

Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.
Paper Structure (14 sections, 3 theorems, 77 equations, 13 figures)

This paper contains 14 sections, 3 theorems, 77 equations, 13 figures.

Table of Contents

  1. Finite-volume approximation of Darcy flow--transport system
  2. Neural basis generation and evaluation
  3. Neural basis expressivity and stability
  4. Geometric interpretation of dual-layer neural basis
  5. Effect of dual-layer neural basis in conditioning
  6. Effect of physics-conforming neural vector bases
  7. Construction of physics-conforming neural vector bases for 3D
  8. Physics-informed neural network (PINN) baseline
  9. NBM--OL implementation details and experiment configurations
  10. Common implementation details
  11. Case-I: permeability-varying steady Darcy flow operator
  12. Case-II: boundary-flux-varying Darcy flow operator
  13. Case-III: boundary-flux/concentration-varying Darcy--transport operator
  14. Geostatistical permeability field generation

Key Result

Lemma 1

Let $V$ be a Hilbert space equipped with norm $\|\cdot\|_V$, and let $a(\cdot,\cdot):V\times V\to\mathbb{R}$ be a continuous bilinear form with continuity constant $C_a>0$ and stability constant $\gamma>0$ (either coercive or inf--sup stable). Let $\{\phi_{\omega}\}_{\omega\sim\pi}\subset V$ denote Let $u\in V$ be the exact solution of and let $u_{N_b}\in V_{N_b}$ denote the least-squares projec

Figures (13)

  • Figure 1: Overview of NBM as a neural PDE solver for coupled multiscale Darcy flow--transport. Left, Physical system. Middle-left, Neural basis approximation for scalar and vector field. Middle-right, Mixed weighted least-squares projection for multiscale Darcy flow. Right, Time marching of Darcy flow and transport.
  • Figure 2: NBM solves advective Darcian system with spectral-like accuracy. a, Homogeneous CO$_2$ storage model. b, Residual $\mathcal{E}_{\mathrm{rel}}$ vs. neural basis size $N_b$. c, Relative $L_2$ errors vs. $N_b$. d, Kolmogorov--Smirnov distance of NBM velocity field vs. $N_b$. e, Refinement of NBM pressure and concentration fields with increasing $N_b$. f, Left to right: ground-truth pressure, followed by absolute error fields (w.r.t. ground truth) for NBM, FVM, and vanilla PINN. g, Left to right: ground-truth velocity magnitude, followed by absolute error fields (w.r.t. ground truth) for NBM, FVM, and vanilla PINN. h, Left to right: ground-truth concentration, followed by absolute error fields (w.r.t. ground truth) for NBM, FVM, and vanilla PINN. i, Relative $L_2$ errors. Note: All results are at 90 days. In f--i, NBM uses $N_b=1000$, and all methods (NBM, FVM, and vanilla PINN) employ a $50\times 50$ spatial resolution. Pressure is reported in $\mathrm{bar}$, velocity in $\mathrm{m\,day^{-1}}$, and concentration in $\mathrm{ppm}$. See Methods for experiment details.
  • Figure 3: NBM remains accurate and robust for multiscale permeability fields. a, Multiscale CO$_2$ storage model. (permeability contrast $502$). b, Residual $\mathcal{E}_{\mathrm{rel}}$ vs. neural basis size $N_b$. c, Relative $L_2$ difference vs. $N_b$. d, K-S distance vs. $N_b$. e, Refinement of NBM pressure and concentration fields with increasing $N_b$. f, Left to right: baseline pressure, velocity magnitude, and concentration. g, Left to right: absolute differences of NBM with energy-consistent weighting, for pressure, velocity magnitude, and concentration. h, Left to right: absolute differences of NBM without energy-consistent weighting, for pressure, velocity magnitude, and concentration. i, Left to right: absolute differences of PINN, for pressure, velocity magnitude, and concentration.s. j, Summary metrics measured against the baseline. k, Manufactured-solution setup (permeability contrast 1078). l, Residual $\mathcal{E}_{\mathrm{rel}}$ vs. $N_b$. m, Relative $L_2$ error vs. $N_b$. n, Energy spectrum $E(k)$ comparison. o, Ground-truth pressure, velocity magnitude, and concentration fields. p, Left to right: absolute errors of NBM, for pressure, velocity magnitude, concentration. q, Left to right: absolute errors of FVM, for pressure, velocity magnitude, concentration. r, Relative $L_2$ errors. s, Comparing metrics vs. permeability contrast. Note: Results in b--j are at 90 days. In f--j and o-s, NBM uses $N_b=1000$, and all methods (NBM, FVM, and vanilla PINN) employ a $50\times 50$ spatial resolution. Pressure is reported in $\mathrm{bar}$, velocity in $\mathrm{m\,day^{-1}}$, and concentration in $\mathrm{ppm}$. See Methods for experiment details.
  • Figure 4: NBM-OL for parametric Darcy--transport systems. Panel I, Parameterization of permeability and boundary value. Panel II, NBM--OL for Darcy flow (per Darcy time step, $\Delta T$). Two networks $\mathrm{MLP}_{p}$ and $\mathrm{MLP}_{q}$ learn the pressure and mass-flux solution coefficients $\boldsymbol{\theta}_{\mathrm{p}}$ and $\boldsymbol{\theta}_{\mathrm{q}}$. Training is self-supervised by $\min \lVert \boldsymbol{R}(\boldsymbol{\theta}_{\mathrm{p}},\boldsymbol{\theta}_{\mathrm{q}}) \rVert_{2}^{2}$ from \ref{['eq:nbm_darcy_picard']}, where $\boldsymbol{R}(\theta_{\mathrm{p}},\theta_{\mathrm{q}})=\mathbf{W}\bigl(\mathbf{A}_{\mathrm{flow}}\,\boldsymbol{\theta}_{p,q}-\mathbf{b}_{\mathrm{flow}}\bigr)$ with time stepping indices omitted. $p^n$ denotes the pressure from the previous Darcy time-step, used for the current update. Panel III, NBM--OL for transport (per transport time-step, $\Delta t$). A dedicated network $\mathrm{MLP}_{c}$ learns concentration coefficients $\boldsymbol{\theta}_{\mathrm{c}}$ by $\min \lVert \boldsymbol{R}(\boldsymbol{\theta}_{\mathrm{c}}) \rVert_{2}^{2}$ from \ref{['eq:nbm_transport_upwind']}, where $\boldsymbol{R}(\theta_{\mathrm{c}})=\mathbf{A}_{\mathrm{up}}\,\boldsymbol{\theta}_{c}-\mathbf{b}_{\mathrm{up}}$ with time stepping indices omitted. $\mathbf{u}^{n+1}$ denotes the velocity after the current Darcy time-step update, computed as $\mathbf{u}^{n+1}=\mathbf{q}^{n+1}/\rho^{n+1}$. Panel IV, Optional POD compression of the solution coefficient snapshots. $\Psi_{p}$, $\Psi_{q}$, and $\Psi_{c}$ are the respective reduced bases.
  • Figure 5: NBM--OL learns parametric Darcy flow operators under varying permeability and boundary flux. a, Problem setup for permeability-varying (kv) Darcy flow operator learning. b, Representative block25 parametrizations. c-d, Left to right: true pressure/velocity magnitude, predicted pressure/velocity magnitude, respective absolute error fields. e, In-distribution generalization, measured by relative $L_2$ errors over 100 random runs. f-h, Out-of-distribution generalization across heterogeneity level, correlation length, and spatial rotation, measured by relative $L_2$ errors over 100 random runs. i, Problem setup for boundary-flux-varying (bv) Darcy flow operator learning. j-k, Left to right: true pressure/velocity magnitude, predicted pressure/velocity magnitude, respective absolute error fields, at 200 days. l, In-distribution generalization, measured by relative $L_2$ errors at 200 days over 100 random runs. m, Out-of-distribution generalization in boundary flux, measured by relative $L_2$ errors at 200 days over 100 random runs. n, Out-of-distribution generalization in time horizon, measured by relative $L_2$ errors at 400 days over 100 random runs. o, Residual--error correlation during training: $\sqrt{\mathcal{E}_{\mathrm{rel}}}$ vs. relative $L_2$ errors. p, Training dynamics measured by normalized $\mathcal{E}_{\mathrm{rel}}$. q, Runtime comparison and speedups of NBM--OL relative to FVM, reported as CPU wall-time statistics in seconds over 100 random runs. Note: Pressure is reported in $\mathrm{bar}$ and velocity in $\mathrm{m\,day^{-1}}$. See Methods for experiment details.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Lemma 1: Probabilistic Céa-type bound for neural basis spaces
  • proof
  • Lemma 2: Residual--error equivalence for the weighted first-order least-squares Darcy problem
  • proof
  • Lemma 3: Residual--error equivalence for upwind control-volume transport problem
  • proof