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

Yuhe Wang; Min Wang

math.NAFeb 19, 2026

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

Authors: Yuhe Wang, Min Wang

Organizations: Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of Houston, Houston, TX 77204, USA

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.

Figures & tables

Fig. 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.

Fig. 2: NBM performance on a homogeneous compressible benchmark. 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 vs. $N_{b}$ . e, Refinement of pressure and concentration solutions with $N_{b}$ . 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 $\mathrm{ppm}$ .

Fig. 3: NBM performance on multiscale benchmarks. 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 $\mathrm{ppm}$ .

Fig. 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, $\Delta T$ ). $\mathrm{MLP}_{p}$ 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 ( 13 ), 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, $\Delta t$ ). $\mathrm{MLP}_{c}$ learns concentration coefficients $\boldsymbol{\theta}_{\mathrm{c}}$ by $\min\lVert\boldsymbol{R}(\boldsymbol{\theta}_{\mathrm{c}})\rVert_{2}^{2}$ from ( 14 ), 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 neural basis coefficients. $\Psi_{p}$ , $\Psi_{q}$ , and $\Psi_{c}$ are the respective reduced POD bases.

Fig. 5: NBM–OL learns parametric Darcy flow operators under varying permeability and boundary flux. a, Problem setup (kv) and base permeability field. 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. f-h, Out-of-distribution generalization across heterogeneity level, correlation length, and spatial rotation. i, Problem setup (bv) and permeability field. j-k, Left to right: true pressure/velocity magnitude, predicted pressure/velocity magnitude, respective absolute error fields, at 200 days. l, In-distribution generalization. m, Out-of-distribution generalization in boundary flux. n, Out-of-distribution generalization in time horizon. 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. Note: Pressure is reported in $\mathrm{bar}$ and velocity in $\mathrm{m\,day^{-1}}$ .

Fig. 6: NBM–OL learns parametric Darcy–transport operators under varying boundary flux and concentration. a, Problem setup for boundary-flux-varying (bv) and injection-concentration-varying (cv) Darcy–transport operator learning. b–c, Left to right: true concentration, predicted concentration with the tracer front overlaid, and the corresponding absolute error; b is at 20 days and c is at 200 days. d, In-distribution generalization, measured by relative $L_{2}$ errors at 200 days over 100 random tests. e, Out-of-distribution generalization in boundary flux and concentration, measured by relative $L_{2}$ errors at 200 days over 100 random runs. f, Out-of-distribution generalization in time horizon, measured by relative $L_{2}$ errors at 400 days over 100 random runs. g, Training dynamics measured by normalized $\mathcal{E}_{\mathrm{rel}}$ . h, Residual–error correlation during training: $\sqrt{\mathcal{E}_{\mathrm{rel}}}$ vs. relative $L_{2}$ errors. i, Runtime comparison and speedups of NBM–OL relative to FVM, reported as CPU wall-time statistics in seconds over 100 random runs. Note: Concentration is reported in $\mathrm{ppm}$ .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. C.1: Geometric illustration of dual-layer neural basis. a, A first-layer hyperplane after activation. b, A second-layer pre-activation field formed by superposing $N_{1}=10$ first-layer features. c, A second-layer pre-activation field formed with $N_{1}=200$ first-layer features. d, Schematic of dual-layer basis generator: input produces first-layer hyperplanes, superpositions form pre-activations fields, and post-activation outputs the final basis functions.

Fig. C.2: Effect of depth in neural basis generation on numerical conditioning. Condition number of the assembled neural projection system versus neural basis size

N_{b}

for single-layer, dual-layer, and triple-layer basis generators (logarithmic

y

-axis).

Fig. D.1: Neural basis expressivity and stability for pressure. a, FVM reference pressure $p$ (bar). b-c, Absolute projection error using dual-layer and single-layer scalar neural basis, respectively d, Relative $L_{2}$ projection error and condition number of the projection system. Note: Pressure is reported in $\mathrm{bar}$ .

Fig. D.2: Neural basis expressivity and stability for mass-flux. a, FVM reference (kg m -2 s -1 ). b-e, Absolute projection error using dual-layer neural vector basis w/ Helmholtz decomposition, dual-layer neural vector basis w/o Helmholtz decomposition, single-layer neural vector basis w/ Helmholtz decomposition, and single-layer neural vector basis w/o Helmholtz decomposition, respectively. f, Relative $L_{2}$ projection errors for $q_{x}$ , $q_{y}$ , and $|\mathbf{q}|$ and condition numbers of the corresponding projection systems. g, Energy spectra $E(k)$ of the reference and the best projected mass-flux fields. Note: Mass-flux is reported in $\mathrm{kg~day^{-1}~m^{-2}}$ .

Fig. D.3: Neural-basis expressivity and stability for concentration. a, FVM reference. b-c, Absolute projection error using dual-layer and single neural scalar basis, respectively. d, Relative

L_{2}

projection error and condition number of the projection system. e, Sharp-front field with zero transition thickness (discontinuous step). f, Slightly smeared front with a 5-cell transition (stabilized profile). g-h, Absolute projection error for sharp-front and smeared-front cases, respectively. i-j, Cell values near sharp-front and smeared-front, respectively. k, Relative

L_{2}

projection errors for the sharp-front and smeared-front fields, and the relative

L_{2}

difference between the two fields (sharp vs. smeared). Note: Concentration is reported in

\mathrm{ppm}

Table E.1: Effect of physics-conforming neural vector bases.

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Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA · Department of Mathematics, University of Houston, TX 77204, USA

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Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.

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