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Solving and Learning Nonlinear PDEs with Gaussian Processes

Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M Stuart

arXiv:2103.12959v2math.NAstat.ML

TL;DR

The paper addresses how to solve nonlinear PDEs and related inverse problems within a unified Gaussian-process framework. It formulates PDE-constrained MAP estimation, reduces it using a representer theorem, and solves the resulting finite-dimensional problem with Gauss–Newton. The framework is theoretically convergent and performs effectively across nonlinear PDE and inverse-problem experiments, though its guarantees depend on kernel suitability and other assumptions.

  • Problem

    The paper seeks a unified kernel and Gaussian-process framework for solving nonlinear PDEs and inverse problems, extending methods previously developed mainly for linear PDEs.

  • Method

    The method approximates PDE solutions as Gaussian-process MAP estimators under collocation constraints, then uses a representer theorem and Gauss–Newton optimization to solve for the solution and inverse-problem parameters.

  • Results

    The method is provably convergent under stated regularity and uniqueness assumptions and demonstrates effective performance on nonlinear PDEs, Burgers’ equation, and inverse problems.

  • Takeaways & Limitations

    The framework provides a common computational and theoretical approach for nonlinear PDE solution and parameter identification using kernel methods.

  • Takeaways & Limitations

    The approach depends on choosing a kernel adapted to the PDE solution, and existing complexity guarantees apply only to restricted functionals and Green’s-function kernels.

Abstract

from arXiv · show

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

1. Introduction.

The paper develops a Gaussian-process and kernel framework for nonlinear PDEs and inverse problems, combining finite-dimensional optimization, convergence guarantees, and practical numerical algorithms.

  • The framework generalizes kernel collocation methods to nonlinear PDEs and inverse problems while connecting to radial-basis and meshless kernel methods.
  • A PDE solution is approximated as a Gaussian-process MAP estimator conditioned on PDE constraints at finitely many collocation points.In the nonlinear case, this estimator is generally not the conditional expectation because the conditioned process is no longer Gaussian.
  • A representer theorem reduces the infinite-dimensional variational problem to finite-dimensional optimization over kernel evaluations and derivative values at collocation points.The resulting minimizer has the form u†(x) = K(x, φ)K(φ, φ)^−1z† and can be obtained from an unconstrained finite-dimensional problem.
  • The method is provably convergent under regularity, compact-embedding, and unique-classical-solution assumptions as the number of collocation points increases.The convergence is pointwise and in H^t(Ω) for any t < s under the stated Sobolev condition.
  • Kernel choice is central: accuracy and convergence depend on adapting the RKHS to the PDE solution, while dense-kernel computational bottlenecks remain relevant.The paper notes that an unsuitable kernel can perform arbitrarily badly and that existing complexity guarantees cover only restricted functionals and kernels.

2. Conditioning GPs on Nonlinear Observations.

The framework connects Gaussian-process conditioning with quadratic Banach-space structure, yielding finite representations for linear observations and nonlinear PDE or inverse-problem constraints.

  • Gaussian-process foundations: The RKHS/GP framework is formulated through Banach spaces with quadratic norms and covariance operators.The associated space coincides with the GP’s Cameron–Martin space and, for covariance kernels, with the kernel’s RKHS.
  • Linear conditioning: Conditioning a canonical GP on linear observations produces a Gaussian distribution with an explicitly represented conditional mean and covariance.Invertibility of the observation covariance matrix Θ yields the finite conditional mean representation.
  • Variational representation: The conditional mean is characterized variationally as the minimum-norm function satisfying the observation constraints, using gamblets as representers.This provides the representer-theorem foundation for the nonlinear PDE framework.
  • Nonlinear observations and inverse problems: For nonlinear inverse problems, the solution and unknown parameter are concatenated, while nonlinear maps encode the forward PDE and observed noisy data.The resulting quadratic optimization problem still admits a finite gamblet expansion.
  • Noise modeling: The observation-noise model assumes i.i.d. noise but extends to correlated noise through a covariance-weighted misfit term.The weighted term uses |Σ^-1/2(G(z) − o)|^2, where Σ is the observation-noise covariance matrix.
  • MAP interpretation: The optimization minimizer is informally interpretable as a GP MAP point conditioned on PDE constraints, but a proof is left for future work.The MAP interpretation is described without equality constraints rather than established formally for the full setting.

3. Solving Nonlinear PDEs.

The method approximates a nonlinear PDE solution by conditioning a Gaussian process at interior and boundary collocation points, then reducing the problem to finite-dimensional optimization.

  • Problem setup: The PDE framework assumes a bounded domain, a nonlinear differential operator, boundary conditions, and a unique strong solution.The solution space is chosen as a sufficiently regular quadratic Banach space, such as a Sobolev space.
  • Problem setup: The proposed approximation conditions a Gaussian process to satisfy the PDE at finitely many collocation points and computes its MAP point.Interior points enforce the PDE, while boundary points enforce boundary data.
  • Constraint formulation: The collocation objective seeks the minimum-norm Cameron–Martin-space element satisfying PDE and boundary constraints at the selected points.Differential operators define the pointwise quantities entering the nonlinear PDE and boundary maps.
  • Constraint formulation: For nonlinear elliptic equations, the framework represents the PDE through linear differential operators combined with algebraic nonlinear maps.The example uses u and ∆u as operator outputs and P(v1, v2) = −v2 + τ(v1).
  • Finite-dimensional reduction: A finite-dimensional representer theorem expresses the minimizer using gamblets, with coefficients obtained by solving a reduced nonlinear optimization problem.The covariance matrix and gamblets can be approximated offline, leaving the coefficients as the numerical optimization variables.
  • Finite-dimensional reduction: The equality-constrained reduced problem is transformed into an unconstrained problem through constraint elimination or relaxation.The numerical algorithms operate on these transformed formulations after the convergence analysis.

3.2. Convergence Theory.

Under regularity, embedding, uniqueness, and collocation-density assumptions, the method’s minimizers converge to the classical PDE solution as the number of collocation points grows.

  • Convergence theorem: The convergence theorem states that minimizers converge to the unique classical solution as the number of collocation points tends to infinity.The result applies when the solution space is compactly embedded in a sufficiently regular space supporting pointwise PDE derivatives.
  • Convergence theorem: The proof obtains convergence through compactness, continuity of the PDE operators, satisfaction of constraints at collocation points, and uniqueness of the PDE solution.Convergence in the regularity space implies convergence in the continuous-function spaces used to evaluate the PDE and boundary conditions.
  • Convergence conditions: The convergence theorem provides no convergence rate and requires the covariance operator or kernel to be adapted so that the true solution belongs to the chosen space.Kernel smoothness can be incompatible with irregular forcing or boundaries, motivating alternative Green’s-function kernels.
  • Convergence conditions: The collocation-point assumptions require the points to become dense in the domain as their number increases.This density supports passage from pointwise constraint satisfaction to the full PDE solution.
  • Constraint handling: Constraint elimination requires finite-dimensional solution maps for selected variables, but variable choice can affect conditioning and the assumption may fail for some nonlinear PDEs.Relaxation introduces a penalty parameter whose vanishing with increasing collocation points restores constraint satisfaction in the limit.

3.4. Implementation.

Implementation uses kernel-based covariance matrices and Gauss–Newton iterations to solve the reduced nonlinear problems, with successive linearized PDE collocation interpretations.

  • Kernel construction: The numerical framework constructs the covariance matrix Θ from kernel evaluations and differential operators applied to its arguments.The matrix dimension is N = MQb + MΩ(Q − Qb), reflecting boundary and interior operator measurements.
  • Gauss–Newton solver: A Gauss–Newton variant updates reduced variables by solving quadratic subproblems formed from linearizations of the nonlinear maps.The gradient is taken with respect to the unconstrained optimization variable, while step sizes may be fixed or adaptive.
  • Gauss–Newton solver: The subproblems can be solved exactly and efficiently, and experiments found rapid convergence using unit step sizes.The reported experiments used αℓ = 1 without requiring adaptive step-size selection.
  • PDE interpretation: The Gauss–Newton sequence coincides with successive kernel-collocation approximations of appropriately linearized PDEs.For the nonlinear elliptic example, linearizing τ produces the corresponding linearized PDE iteration.
  • Computational cost: The primary computational cost is approximating and factoring Θ^-1, for which the paper uses standard O(N^3) factorization algorithms.More efficient factorizations and approximations are available from GP regression and numerical-analysis literature but are not used here.
  • Computational cost: Gamblets can be approximated with compactly supported functions in complexity O(N log^(2d+1)(N/ε)) under specific delta-functional and elliptic-Green’s-function assumptions.The cited complexity-versus-accuracy guarantees have not been established for the paper’s full generality of functionals and kernels.

3.5. Numerical Experiments for Nonlinear PDEs.

The method is evaluated on three nonlinear PDEs using automatic differentiation, random collocation points, and Gauss–Newton optimization. Accuracy improves with more collocation points, while convergence typically requires only a few iterations but depends on problem features and numerical choices.

  • Experimental setup: Experiments solve nonlinear elliptic, Burgers, and regularized Eikonal equations using random collocation points and Gauss–Newton optimization.Automatic differentiation forms derivative-dependent kernel matrices and optimizes the loss function.
  • Nonlinear elliptic PDE: 3 steps sufficed for convergence in the nonlinear elliptic experiment with M = 1024 collocation points.Errors were fairly uniform spatially, with larger errors near the boundary.
  • Nonlinear elliptic PDE: Elimination and relaxation produced only mildly different solution errors for the nonlinear elliptic PDE, although elimination appeared more efficient.The comparison used M = 300, 600, 1200, 2400 and results averaged over 10 random-collocation realizations.
  • Burgers’ equation: 10 steps sufficed for Burgers’ equation, with maximum errors occurring near the viscous shock at time 1.Time slices at t = 0.2, 0.5, 0.8 showed that the method captured the shock’s location and shape.
  • Burgers’ equation: Smaller viscosity can significantly reduce Burgers’ accuracy because sharper shocks require kernels and collocation points suited to shorter length and time scales.This links accuracy to the viscosity parameter ν and the resulting shock sharpness.
  • Regularized Eikonal equation: 6 iterations sufficed for the regularized Eikonal equation, and accuracy improved clearly as the number of collocation points increased.Errors were larger in the middle of the domain and near the boundary.

4. Solving Inverse Problems.

The framework extends to inverse PDE problems by jointly estimating the solution and unknown parameters under PDE constraints. A Darcy-flow experiment shows that both recovered quantities capture the truth’s shape despite limited observations.

  • Framework: The inverse-problem framework identifies PDE parameters from limited, noisy observations while estimating the PDE solution simultaneously.The parameter and solution are modelled with Gaussian processes and conditioned to satisfy the PDE at collocation points.
  • Framework: The inverse formulation supports independent or correlated Gaussian-process priors for the solution and parameter.A product-space prior introduces correlations while retaining the same PDE constraints.
  • Darcy-flow inverse problem: The Darcy-flow inverse problem targets the coefficient a from noisy pointwise observations of u at selected points.The PDE representation uses linear operators for u, ∇u, ∆u, a, and ∇a, with nonlinear map P defined explicitly.
  • Optimization: The representer-theorem reduction converts the inverse problem into finite-dimensional optimization over measurement and PDE functionals, under invertibility assumptions.Corollary 4.4 applies when the covariance operators and associated matrices are invertible.
  • Scope and assumptions: Limited and noisy data generally prevent exact recovery, so kernel choices should reflect prior assumptions about the solution and parameter.The covariance operators must also provide sufficient regularity for the pointwise constraints.

5. Concluding Remarks.

The paper establishes convergence and error-estimation results for its nonlinear PDE framework, while identifying conditions affecting uniqueness, optimality, and kernel suitability. It also notes that kernel selection remains challenging for nonlinear PDEs and irregular solutions.

  • Theoretical guarantees: The nonlinear representer theorem reduces the MAP estimation problem to finite dimensions and supports convergence toward the truth as collocation points become dense.The convergence result requires regularity assumptions.
  • Limitations: Uniqueness of the minimizer is not guaranteed in general, and the convergence theorem does not provide convergence rates.Nonlinearity makes the constrained optimization problem nonconvex; additional assumptions can recover uniqueness.
  • Theoretical guarantees: Under condition (5.1), the minimizer u† is unique and has an optimal-recovery interpretation.Condition (5.1) constrains the nonlinear manifold relative to a tangent hyperplane; it is automatically satisfied with equality for affine F.
  • Error estimates: The framework provides rigorous error estimates by connecting conditional GP covariance with pointwise errors of the conditional MAP estimator.For suitable compact embeddings, the associated approximation terms converge to zero as collocation points become dense.
  • Error estimates: The strong approximation error is bounded by a kernel-covariance approximation term plus an additional error term.The bound is expressed as ∥u† − u⋆∥ ≤ inf_{φ∈spanφ} ∥u⋆−Kφ∥ + ϵ.
  • Limitations: Accuracy depends critically on choosing a kernel whose RKHS contains the PDE solution, while suitable kernels can be difficult to select for nonlinear PDEs with irregular solutions.The framework permits kernel learning through MLE, MAP estimation, or cross-validation.
  • Future directions: The framework can potentially be generalized from Gaussian priors to nonlinear transformations of Gaussian measures when the transformed process retains sufficiently regular derivatives.This is presented as a possible direction rather than an established result.

Appendix A. Diagonal Regularization of Kernel Matrices.

The appendix interprets diagonal kernel-matrix regularization as a nugget, relating it to small Gaussian observation noise and extending the terminology to perturbed matrices used here.

  • Nugget interpretation: A nugget is a small diagonal regularization added to a kernel matrix, interpreted in GP regression as small Gaussian noise on observations.The paper uses the same terminology for slight perturbations of Θ and eΘ.

A.1. An Adaptive Nugget Term.

The adaptive nugget addresses ill-conditioning in derivative-structured kernel matrices by scaling diagonal blocks according to their relative magnitudes. This improves numerical stability while managing the accuracy tradeoff.

  • Motivation: Ill-conditioning in Θ requires regularization, but a uniform ηI can be suboptimal because derivative blocks may have widely different spectral scales.Regularization can introduce an accuracy floor, creating an accuracy-stability tradeoff.
  • Construction: The adaptive nugget ηR reweights each diagonal block using the trace ratio between that block and the pointwise-measurement block.A single global nugget parameter η controls the resulting block-scaled regularization.
  • Computational implementation: With a suitable η, the adaptive nugget supports fast, stable Gauss–Newton convergence and permits offline Cholesky factorization for repeated iterations.Each iteration then uses back-substitution with the precomputed factors.
  • Example: For M = 1024 collocation points, the elliptic-PDE experiment had a trace ratio of order 4000 between pointwise and Laplacian blocks.This makes the distinction between uniform and adaptive identity weighting substantial.

A.2. Choice of Nugget Terms for the Nonlinear Elliptic PDE.

The nonlinear elliptic PDE experiments compare standard ηI regularization with the adaptive ηR nugget using converged Gauss–Newton solutions and averaged error measurements.

  • Experimental settings: The experiments use adaptive nugget parameters η = 10^-13 for Figure 2 and η = 10^-12 for Table 1.These parameter choices are tied to the reported experimental results.
  • Comparison: The comparison evaluates L2 and L∞ errors for standard ηI and adaptive ηR regularization across 10 random collocation-point samplings.Cholesky factorization and subsequent Gauss–Newton iterations are used for both regularization choices.

A.3. Choice of Nugget Terms for Burger’s Equation.

The experiments primarily used adaptive nugget regularization, while Table 4 compares standard and adaptive nugget terms under fixed nonlinear-elliptic settings.

  • The numerical experiments primarily used the adaptive nugget term described in Appendix A.1.
  • Table 4 compares solution errors for standard and adaptive nugget terms in the nonlinear elliptic PDE experiment.The comparison used M = 1024 and MΩ= 900, with results averaged over 10 random collocation-point realizations.
  • The Table 4 experiments used a Gaussian kernel with lengthscale parameter σ = 0.2 and allowed at most five Gauss-Newton iterations.

A.4. Choice of Nugget Terms for the Eikonal Equation.

For the Eikonal-equation convergence analysis, the nugget parameter was varied with the number of collocation points, improving computational efficiency by avoiding slower convergence at smaller M.

  • The Eikonal experiments used the adaptive nugget strategy described in Appendix A.1.
  • The convergence analysis varied η with M, using η = 10−5 for M ≤1200 and η = 10−10 for M ≥2400.
  • Smaller η values for smaller M caused slower Gauss-Newton convergence, so varying η with M improved framework efficiency.

A.5. Choice of Nugget Terms for Darcy Flow.

In the Darcy-flow inverse-problem setting, both dense, ill-conditioned matrices were regularized independently with adaptive nuggets, using equal global parameters in Figure 5.

  • Both Θ and eΘ are dense and ill-conditioned in the inverse-problem setting, motivating nugget regularization.
  • Adaptive nuggets were added independently to Θ and eΘ using re-weighted identity matrices and the strategy from Appendix A.1.
  • The Figure 5 experiments used η = ˜η = 10−5 for the two regularized matrices.
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