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Physics-informed learning of governing equations from scarce data

Zhao Chen, Yang Liu, Hao Sun

arXiv:2005.03448v3cs.LGphysics.comp-phphysics.data-anstat.ML

TL;DR

Discovering governing PDEs from scarce, noisy measurements remains challenging because existing sparsity-promoting methods scale poorly under these conditions. This paper combines deep learning, automatic differentiation, and sparse regression to identify PDEs, demonstrating accurate and robust discovery across several systems and levels of data scarcity and noise.

  • Problem

    Discovering governing PDEs from scarce and noisy data is challenging, while existing sparsity-promoting methods scale poorly with data noise and scarcity.

    Source: DISCUSSION · p. 12

  • Method

    The framework combines deep neural networks, automatic differentiation, and ℓ0 sparse regression, using alternating direction optimization to identify parsimonious PDEs.

    Source: PiDL with Sparse Regression for PDE… · p. 5, DISCUSSION · p. 12

  • Results

    The method accurately discovers Schrödinger, Navier–Stokes, and λ-ω reaction-diffusion equations and shows robustness to measurement scarcity and noise.

    Source: Discovery of Benchmark PDEs with… · p. 6

  • Takeaways & Limitations

    The framework shows potential for discovering parsimonious closed-form PDEs from scarce and noisy data in scientific and real-world applications.

    Source: DISCUSSION · p. 12

    Applying the fully connected network to higher-dimensional systems can create computational bottlenecks and optimization challenges because satisfactory accuracy requires many collocation points.

    Source: DISCUSSION · p. 14

Abstract

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Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel physics-informed deep learning framework to discover governing partial differential equations (PDEs) from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to (1) approximate the solution of system variables, (2) compute essential derivatives, as well as (3) identify the key derivative terms and parameters that form the structure and explicit expression of the PDEs. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of PDE systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.

RESULTS

The results examine PiDL’s discovery of governing PDEs from scarce, noisy measurements across canonical benchmarks, multiple initial- and boundary-condition datasets, and cell-migration experiments. In the cell-migration application, the identified PDE is validated against observed cell-density evolution, and the estimated migration and proliferation rates generally increase with cell number.

Discovery of Benchmark PDEs with Single Dataset

  • PiDL recovered closed-form Burgers’, KS, nonlinear Schrödinger, Navier–Stokes, and λ-ω reaction-diffusion equations from sparse, noisy single-IBC data, with predicted responses close to exact solutions.The reported results include 10%-noise cases; the table summarizes discovery accuracy across canonical models, while Figs. 2–3 show coefficient evolution and predicted responses.Source: Table 1 · p. 5, Figure 2 · p. 6, Discovery of Benchmark PDEs with… · p. 6

PiDL with Sparse Regression for PDE Discovery

  • A DNN approximates latent solutions and derivatives for a candidate-term library, while sparse PDE constraints regularize that modeling; alternating direction optimization (ADO) finds parsimonious equations.The method uses ℓ1-regularized pretraining to initialize ADO, which addresses the intractability of direct optimization with ℓ0 regularization.Source: PiDL with Sparse Regression for PDE… · pp. 4–5

Discovery of Benchmark PDEs with Single Dataset

  • Burgers’ discovery achieved 1.2% coefficient error and 2.02% full-field prediction error; the method also identified an unknown sin(x) sin(t) source alongside the PDE.For KS, coefficient error was around 0.7% and relative ℓ2 prediction error was 1.87%; for nonlinear Schrödinger, the corresponding errors were about 4.14% and 1%.Source: Discovery of Benchmark PDEs with… · pp. 7–8

Discovery of PDEs with Multiple Independent Datasets

  • Across three noisy IBC datasets, the root-branch network recovered Burgers’ ut = −1.006uux + 0.0039uxx and achieved a stacked full-field ℓ2 error of 2.24%.For the Fitzhugh-Nagumo system, the equation forms and active terms were recovered, but γv was less precisely identified; the stacked full-field ℓ2 error was 5.04%.Source: Discovery of PDEs with Multiple… · pp. 9–11
  • The multiple-dataset Fitzhugh-Nagumo case showed that scarce, noisy measurements and sharp propagating fronts can reduce solution-approximation and coefficient precision even when the PDE form is recovered.The authors note that increasing spatiotemporal measurement resolution could alleviate the solution-approximation issue.Source: Discovery of PDEs with Multiple… · pp. 10–11

Experimental Discovery of Cell Migration and Proliferation

  • The experiments used mean data from three replicates across four initial cell counts, from 14,000 to 20,000, to discover a sparse PDE for cell density.The model assumed diffusion and reaction terms, with eight candidate terms for the nonlinear reaction functional.Source: Experimental Discovery of Cell… · p. 12
  • The discovered PDEs share the form ρ_t = γρ_xx + λ1ρ + λ2ρ^2, matching the Fisher-Kolmogorov model.The identified migration and proliferation rates generally increase with the number of cells.Source: Experimental Discovery of Cell… · p. 12
  • Predictions at 12–48 h agree satisfactorily with measurements using the 0 h density as the initial condition and Neumann boundary conditions.The passage notes that extremely scarce and noisy data challenge other methods such as SINDy to produce a reasonable discovery.Source: Experimental Discovery of Cell… · p. 12

DISCUSSION

The discussion presents PiDL as a framework for discovering closed-form governing PDEs from scarce and noisy data, combining neural solution modeling with sparse equation representation. It reports accurate equation discovery across nonlinear spatiotemporal systems and outlines computational and scalability limitations.

DISCUSSION

  • The method couples DNN solution approximation and automatic differentiation with ℓ0 sparse regression, while collocation points support generalization from scarce, noisy measurements.Its architecture accommodates independent datasets under different initial/boundary conditions, and alternating-direction optimization jointly trains the DNN and sparse coefficients.Source: DISCUSSION · pp. 12–13
  • For cell migration datasets of 14,000–20,000 cells, the discovered PDEs use active terms and are simulated against measured cell densities under specified initial and Neumann boundary conditions.The passage identifies the active terms as t∆ρ, ρ, and ρ2u; the 0h measurement is used as the initial condition.Source: DISCUSSION · p. 13, Figure 6 · p. 13
  • PiDL accurately discovers governing PDE forms from scarce, noisy measurements across nonlinear spatiotemporal systems, including numerical and experimental cases.The demonstrations cover single or multiple datasets, including chaotic, shock, and propagating-front dynamics.Source: DISCUSSION · p. 13
  • Fully connected DNNs pose computational and optimization bottlenecks for higher-dimensional systems because accurate modeling can require many collocation points.The discussion suggests discrete spatiotemporal networks such as ConvLSTM as a potential future remedy.Source: DISCUSSION · p. 14
  • The root-branch architecture may scale poorly with many independent datasets, as trainable variables, collocation requirements, and memory grow linearly with dataset count.The discussion identifies multi-GPU parallelization or a parametric DNN for known, parameterizable initial/boundary conditions as potential approaches.Source: DISCUSSION · p. 14

METHOD

PiDL combines neural solution approximation, physics embedding, automatic differentiation, and sparse regression to discover PDE structure and coefficients from scarce, noisy data. Its architecture, physics-constrained sparse loss, and alternating optimization support shared-law discovery across different initial and boundary conditions.

METHOD

  • PiDL integrates deep networks, physics embedding, automatic differentiation, and sparse regression to approximate solutions, compute derivatives, and identify PDE terms and coefficients from scarce, noisy data.The approach is designed to account for different initial and boundary conditions.Source: METHOD · p. 14

Network Architecture

  • Single datasets use a dense DNN; multiple IBC datasets use a shared-root, branched network whose root learns common patterns and branches capture IBC-specific responses.Tanh or, for periodic responses, sin activations are used for high-order differentiation; the unified architecture supports solutions satisfying a common governing PDE.Source: Network Architecture · pp. 14–15

Physics-constrained Sparsity-regularized Loss Function

  • The loss combines data fit, PDE-residual constraints at randomly sampled collocation points, and ℓ0 coefficient regularization; concatenated residual libraries enforce a shared law across datasets.Data loss fits the measured solutions, physics loss constrains them through PDE reconstruction, and ℓ0 regularization promotes sparse PDE representations.Source: Physics-constrained… · p. 15

Alternating Direction Optimization

  • To handle the intractable ℓ0 objective, ADO alternates STRidge coefficient updates with DNN training using Adam plus L-BFGS; ℓ1 relaxation risks false positives [43].ℓ1-regularized pre-training can provide an initial guess and accelerate ADO, while optional post-training can fine-tune the discovered coefficients and improve accuracy.Source: Alternating Direction Optimization · pp. 15–16

Supplementary information: The supplementary information is attached. … B.1.1 Burgers’ equation

The supplementary material details the PiDL algorithm and its examples; for Burgers’ equation, it reports accurate equation identification and full-field prediction despite sparse measurements.

Supplementary information: The supplementary information is attached.

  • The supplementary document provides algorithm details, examples, and discussion of technical challenges in discovering closed-form PDEs from scarce, noisy data.Source: Supplementary information: The… · p. 24

A Alternating Direction Optimization (ADO): Algorithm

  • ADO alternates neural-network training with STRidge, which applies sequential thresholding as a proxy for ℓ0 regularization to prune small coefficients.The algorithm describes STRidge as sequential thresholding followed by ridge regression on retained coefficients.Source: A Alternating Direction Optimization… · pp. 24, 26

B Examples

  • The examples assess discovery across canonical PDEs, multiple initial and boundary conditions, noisy data, and experimental cell-migration measurements, with SINDy also used for comparison.The systems include Burgers’, Kuramoto-Sivashinsky, nonlinear Schrödinger, Navier-Stokes, and λ-ω reaction-diffusion equations, plus multi-dataset Burgers and Fitzhugh-Nagumo cases.Source: B Examples · p. 26

B.1.1 Burgers’ equation

  • Burgers’ data used five sensors, 1.6 × 10^5 collocation points, and 16 candidate functions to reconstruct the equation.The source dataset had ν = 0.1, 256 spatial points, 101 time steps, and a Gaussian initial condition; the network had eight hidden layers of width 20.Source: §B.1.1 · pp. 26–27
  • For 10%-noise Burgers data, PiDL identified ut = −0.997uux + 0.098uxx with 1.15±1.20% identification error and 2.02% full-field ℓ2 error.The measurements used only 1.95% of the dataset.Source: §B.1.1 · p. 27

B.1.2 Kuramoto-Sivashinsky equation

PiDL discovers the 1D Kuramoto–Sivashinsky equation from sparse, noisy measurements despite the system’s chaotic dynamics. It recovers the equation’s active terms and predicts the full-field response with low error.

B.1.2 Kuramoto-Sivashinsky equation

  • Reverse diffusion in the KS equation leads to blowup, while its fourth-order derivative introduces chaotic patterns that challenge high-order derivative estimation from poorly measured data.The system evolves from a smooth initial condition to an unstable laminar state under highly nonlinear terms.Source: §B.1.2 · p. 27
  • The experiment used 320 of 1,024 spatial nodes over 101 time steps, 2 × 10^4 collocation points, and a library of 36 candidate functions.The measurements comprise about 12.3% of the original dataset; collocation points were Sobol-sampled in the spatiotemporal domain for the residual physics loss.Source: §B.1.2 · p. 27
  • With 10% noise, PiDL recovered u_t = -0.992u u_x - 0.990u_xx - 0.991u_xxxx with 0.71±0.06% average relative coefficient error; SINDy methods failed on very coarse, noisy measurements.The reported coefficient error is averaged over all nonzero elements in Λ.Source: §B.1.2 · p. 27
  • The trained PiDL predicts the full-field wave with 1.87% relative ℓ2 error, and its spatial and temporal snapshots match the ground truth.The comparison is reported against the exact solution.Source: §B.1.2 · p. 28

B.1.3 Nonlinear Schrödinger equation

The framework discovers a closed-form nonlinear Schrödinger equation from scarce, noisy complex-valued data, with coefficient and full-field predictions close to the simulated solution.

B.1.3 Nonlinear Schrödinger equation

  • With a Gaussian initial condition, the study sampled 256 spatial sensors over 375 time instants from a 512-point, 501-step grid, using 37.5% of the data.Source: §B.1.3 · p. 28
  • The method predicts real and imaginary components separately, recombines them into complex u, and uses magnitude rather than ℓ2 norm for residual physics loss to avoid complex gradients.Source: §B.1.3 · p. 29
  • A 40-function library spans polynomial and absolute-value terms, derivatives, and their combinations; the fully connected network has 8 hidden layers of width 40.Source: §B.1.3 · p. 29
  • Training used 1.6 × 10^5 Adam pre-training epochs with L-BFGS tuning, followed by 30 ADO iterations alternating DNN training with STRidge.Each ADO iteration used 1 × 10^3 Adam epochs and up to 4 × 10^3 L-BFGS epochs.Source: §B.1.3 · p. 29
  • Under 10% noise, the model identified i u_t = -0.490u_xx - 0.974|u|^2u, with 2.31 ± 0.28% average relative nonzero-coefficient error and about 1% full-field relative ℓ2 error.The sparse coefficients converged to their actual values, and predicted real and imaginary fields closely matched the exact solution.Source: §B.1.3 · pp. 29–30

B.1.4 Navier-Stokes equation

The framework identifies the 2D cylinder-flow Navier–Stokes vorticity equation from sparse, noisy measurements and predicts the full-field vorticity response with low error.

B.1.4 Navier-Stokes equation

  • Measurements covered 500 random spatial locations over 60 time steps—0.22% of the dataset and one-tenth of the data used in —with 10% noise.The candidate library contained 60 components, and the latent network output comprised u, v, and w.Source: §B.1.4 · p. 30
  • 1.40 ± 1.83% aggregated relative identification error accompanied recovery of the vorticity equation with coefficients −0.999, −0.994, 0.010, and 0.010.The discovered equation is wt = −0.999uwx − 0.994vwy + 0.010wxx + 0.010wyy.Source: §B.1.4 · p. 30
  • 2.57% was the ℓ2 error of the predicted full-field vorticity response, with snapshots at five times compared against the exact solution.The snapshot responses were not used to train the network.Source: §B.1.4 · p. 30
  • The coefficients converged quickly to the correct values with precise sparsity after the first ADO iteration.The residual physics loss used 6 × 10^4 randomly sampled Sobol collocation points.Source: §B.1.4 · p. 30

B.1.5 λ-ω type Reaction-Diffusion equations

The section tests physics-informed discovery on a two-variable λ-ω reaction-diffusion system using only 0.29% of its total data and measurements with 10% noise. It reports recovery of the governing equations and accurate predicted responses.

B.1.5 λ-ω type Reaction-Diffusion equations

  • The coupled equations evolve u and v through second-order spatial diffusion and cubic nonlinear reaction terms.The stated system contains 0.1 diffusion coefficients and the nonlinear terms uv², u³, v³, and u²v in both equations.Source: §B.1.5 · pp. 31, 34
  • The λ-ω system describes local reactive transformation and global diffusion, and this example exhibits spiral waves under periodic boundary conditions.The binomial system is also called an activator-inhibitor system: one state variable encourages both states’ increase while the other inhibits their growth.Source: §B.1.5 · pp. 32, 34
  • 4.78±3.66% was the average relative error for all non-zero coefficients, and the relative full-field ℓ2 error of the prediction was about 2.21%.The sparse terms and associated coefficients were recovered from 110 candidate functions; predicted snapshots closely matched the ground truth.Source: Figure S.6 · p. 33, §B.1.5 · p. 34
  • The test used 2,500 fixed sensors at 15 sampled time steps—0.29% of total data—and 10^5 collocation points to discover two PDEs from 110 candidates.The 256×256 spatial domain had 201 time steps; the method added post-training of the DNN and discovered coefficients after the ADO stage.Source: §B.1.5 · p. 34

B.2 Comparison with SINDy · B.3 Discovery of PDEs with Multiple Independent Datasets

Across five PDEs, PiDL correctly recovers closed-form equations across varying data sizes and noise levels, while PDE-FIND’s performance depends on measurement quality and quantity. PiDL combines neural representation learning, automatic differentiation, sparse regression, and collocation points, at higher computational cost.

B.2 Comparison with SINDy

  • PiDL correctly uncovers closed-form PDEs across all tested data-size and noise conditions, while PDE-FIND fails as scarcity or noise increases.The comparison covers five PDEs; Table S.1 reports sparse-coefficient discovery errors, marking incorrect term identification as NA.Source: §B.2 · p. 34, Table S.1 · p. 35
  • PDE-FIND achieves negligible error with large, clean measurement data but eventually fails when data become scarcer or noisier.The paper characterizes PDE-FIND as relying on strict requirements for measurement quality and quantity.Source: §B.2 · p. 34
  • The comparison tests five PDEs at different data sizes and noise levels, measuring average relative error in identified nonzero coefficients against ground truth.Incorrect identification of sparse PDE coefficients is marked NA.Source: §B.2 · pp. 34–35
  • PiDL combines deep neural networks for nonlinear representation learning, automatic differentiation for derivative calculation, and ℓ0 sparse regression to identify PDEs.Source: §B.2 · p. 35
  • Collocation points provide additional pseudo-datasets that indirectly compensate for scarce measurements and enrich constraints for constructing closed-form PDEs.Source: §B.2 · p. 35
  • PiDL is much more computationally costly than PDE-PIND, primarily because it requires training deep neural networks.The authors say the cost can be managed with parallel computing on a powerful GPU platform.Source: §B.2 · p. 35

B.3.1 Burgers’ equation with shock behavior

The framework discovers the shock-forming Burgers’ equation from three noisy datasets with different initial/boundary conditions. A shared-root, separate-branch network combines their information to identify one PDE and reproduce the distinct responses.

B.3.1 Burgers’ equation with shock behavior

  • With diffusion coefficient 0.01/π ≈ 0.0032, shock formation creates a compact region of sharp gradients that challenges numerical resolution and DNN approximation.The three datasets use distinct initial/boundary conditions, including a Gaussian initial condition and nonzero boundary values in one case.Source: §B.3.1 · pp. 35–36
  • Each dataset has 30 randomly placed sensors measured at 500 time instants, or 7.5% of grid points, and all measurements include 10% Gaussian noise.A denser sensor grid than in the earlier Burgers’ example was needed to capture shock behavior.Source: §B.3.1 · p. 36
  • The discovered PDE is u_t = -1.006u u_x + 0.0039u_xx, with a stacked full-field ℓ2 error of 2.24%.ADO retained the correct u u_x and u_xx terms and eliminated redundant candidates, accurately identifying the small viscosity coefficient.Source: §B.3.1 · p. 36

B.3.2 Fitzhugh-Nagumo type of Reaction-Diffusion equations

With three scarce, noisy measurement datasets, the framework recovered the exact form of the Fitzhugh–Nagumo reaction-diffusion equations, although some coefficient and solution estimates remained imperfect.

B.3.2 Fitzhugh-Nagumo type of Reaction-Diffusion equations

  • Each of three datasets contained 31 low-resolution 31 × 31 snapshots from t = 7.18–36, sampled from synthetic data with 10% noise and three random initial conditions.The system models two interacting components with diffusion coefficients γu = 1 and γv = 100 and reaction coefficients α = 0.01 and β = 0.25.Source: §B.3.2 · p. 37
  • With diffusion terms assumed known but their coefficients unknown, the method searched a 72-term library to reconstruct nonlinear reactions and constrained γu and γv to positive bounded ranges.The library included up-to-third-order polynomials and derivative products; the constant term was rescaled to increase sensitivity to the small stimulus.Source: §B.3.2 · p. 38
  • The exact PDE form and all active terms, including the unknown stimulus, were recovered; nonzero-coefficient error was 11.72±8.34%, and stacked full-field ℓ2 error was 5.04%.The diffusion coefficient γv was an exception to the generally close coefficient estimates and appeared less sensitive.Source: §B.3.2 · p. 38
  • Scarce noisy data challenged the network’s approximation of sharp propagating fronts, reducing discovery precision; increasing measurement resolution can alleviate this even with 10% noise.The authors prioritize recovering PDE form because coefficients can be calibrated when more data become available.Source: §B.3.2 · p. 38

B.4 Experimental Discovery of Cell Migration and Proliferation · C Discussion · C.1 Selection of candidate functions

From sparse cell-density measurements, the framework recovers a diffusion–reaction PDE that predicts cell-density evolution across four initial cell counts. The discussion highlights candidate-library design as a balance between discovering essential terms and keeping the search tractable.

B.4 Experimental Discovery of Cell Migration and Proliferation

  • The study uses 1D cell-density profiles extracted from imaging at five times from 0h to 48h, with four initial cell counts and mean datasets from three replicates.Each mean dataset contains 38 × 5 measurement points; the experiments span initial counts of 14,000, 16,000, 18,000, and 20,000 cells.Source: §B.4 · pp. 38, 40
  • Assuming diffusion and reaction dynamics, the model uses ρ_t = γρ_xx + F(ρ), with eight candidate reaction terms and nine trainable coefficients.The candidate terms include 1, ρ, ρ^2, ρ^3, ρ_x, ρρ_x, ρ^2ρ_x, and ρ^3ρ_x.Source: §B.4 · p. 40
  • Training combines a three-layer tanh network, 10^4 collocation pairs, and five ADO iterations with STRidge, which prune redundant terms before post-training refines active coefficients.A softplus output enforces positive cell density, while sigmoid and tanh transforms constrain coefficient magnitudes.Source: §B.4 · p. 40

C Discussion

  • The discussion reviews features, influence factors, limitations, and potential future work for data-driven PDE discovery with PiDL.This section frames the subsequent discussion of candidate-function selection.Source: C Discussion · p. 40

C.1 Selection of candidate functions

  • Candidate libraries should be diverse yet manageable; domain knowledge can constrain the search, while very large libraries risk rank deficiency, poor conditioning, and greater computational complexity.Balancing these concerns and identifying mathematical principles for efficient library construction remain open problems.Source: §C.1 · pp. 40–42

B.4 Experimental Discovery of Cell Migration and Proliferation

  • Across 14,000–20,000 cells, the discovered PDEs share the Fisher-Kolmogorov form and predict density evolution in satisfactory agreement with measurements.Identified migration and proliferation rates generally increase with cell number; simulations use the 0h measurement as the initial condition and Neumann boundary conditions.Source: §B.4 · p. 40, Figure S.9 · p. 41

C.1 Selection of candidate functions

  • Omitting essential functions can produce false-positive parsimonious PDEs; one alternative is to start with rich polynomial terms, then use Taylor-series analysis and power-law classes to infer simpler forms and refine the library.The proposed analysis can inform redesign and enrichment of candidate functions for potentially improved discovery.Source: §C.1 · p. 42

C.2 Noisy measurements and collocation points

In Burgers’ equation tests, PiDL’s discovery accuracy depends on measurement noise and data quantity, while global collocation points help it learn from sparse measurements and generalize beyond the observed region.

C.2 Noisy measurements and collocation points

  • More measurement points compensate for noise, while at least 2×10^4 collocation points maintained satisfactory accuracy up to 20% noise; under 40% noise, PiDL remained robust with at least 8×10^4 collocation points.The collocation-point sensitivity test fixed measurement points at 1×10^3; the measurement-count test fixed collocation points at 1.6×10^5.Source: §C.2 · pp. 42–43
  • With global collocation points, PiDL discovered the PDE despite 10% measurement noise; without collocation, response errors exceeded 50% and the PDE was misidentified, while doubling clean measurements still left extrapolation poor.The extrapolation test used measurements from an enclosed region and collocation points sampled across the full spatiotemporal field.Source: Table S.2 · p. 43, §C.2 · pp. 43–44
  • Collocation points need not correlate with measurements: Sobol or Latin hypercube sampling can distribute them across the domain, though large samples improve generalization at added computational and memory cost.The authors suggest using more than 10^4 collocation samples in practice, depending on problem complexity, dimension, and available memory.Source: §C.2 · p. 43
  • The Burgers’ equation extrapolation study used 30 randomly selected fixed sensors in an enclosed area, yielding 1.5 × 10^3 measurements, and sampled 8 × 10^4 full-field collocation points to evaluate residual physics loss.Four cases demonstrated the function of collocation points with measurements sampled in the enclosed area.Source: §C.2 · p. 43

C.3 Simultaneous identification of unknown source term

The framework incorporates source candidates into the PDE library to identify the governing equation and reconstruct an unmeasured source simultaneously; in a sparse, noisy Burgers test, it recovered both and extrapolated the solution field.

C.3 Simultaneous identification of unknown source term

  • Adding source candidate functions to the library enables simultaneous sparse identification of the PDE and reconstruction of the unknown source.Source: §C.3 · p. 44
  • The test used 20 randomly selected fixed sensors over 50 time steps on a 201-by-101 space-time grid; the source was unmeasured.Source: §C.3 · p. 44
  • With 4.9% subsampled responses and 10% noise, nonzero-coefficient relative error averaged 4.39% ± 7.03% and full-field ℓ2 error was 13.8%, with errors concentrated near boundaries.Source: §C.3 · pp. 44–45, Figure S.12 · p. 45
  • The recovered Burgers model was u_t + 1.002u u_x - 0.088u_xx = 0.995 sin(x)sin(t), accurately identifying the PDE terms and source coefficient.Source: §C.3 · p. 44, Figure S.12 · p. 45
  • For a source with completely unknown form, constructing a sufficiently diverse candidate library may require a very large search space and add computational complexity.Source: §C.3 · p. 45

C.4 Other network architecture

The fully connected DNN supports analytical PDE-derivative approximation through automatic differentiation, but higher-dimensional modeling poses computational and optimization challenges; discrete architectures may help if their differentiators are carefully designed.

C.4 Other network architecture

  • Fully connected DNNs enable analytical PDE-derivative approximation via automatic differentiation, but modeling higher-dimensional systems creates computational bottlenecks and optimization challenges.The passage gives long-short-term response evolution in a 3D domain as an example.Source: §C.4 · p. 45
  • Discrete architectures such as ConvLSTM may help address these challenges, but require carefully designed spatiotemporal differentiators for high-order derivatives and irregular domains.The authors say they will demonstrate this potential in future work.Source: §C.4 · pp. 45–46

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