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Where DEQN-JAX sits among the methods you already use

DEQN-JAX is a global solver for recursive economic equilibria. It approximates the decision rule π(s) with a neural network — the role Chebyshev polynomials or splines play in a projection method — and drives the Euler / FOC / market-clearing residuals to zero in expectation over next-period shocks, on the simulated ergodic set. This page is the decision aid: find the row of the method table that is your model, and see when DEQN earns its place next to the tools you already trust.

Status: alpha (v0.2.0)

The validated stack is small: adam + an MLP (or LinearPlusMLP) + an MSE residual + antithetic Monte-Carlo (or Gauss–Hermite) expectations. DEQN is a solver, not an estimator, and not a replacement for Dynare — it composes with it. The two honest limits are stated plainly below, never buried.

Pick the row that matches your model

Same target across the whole table: a decision rule π(s) that zeroes the equilibrium residuals. The methods differ in what approximates π, where it stays accurate, and how far it scales in the state dimension.

Method Approximates π(s) by Accurate where State-dim ceiling Reach for it when
Perturbation (Dynare) Taylor expansion around the steady state a neighborhood of the SS effectively unlimited linear / near-linear models, IRFs, the published baseline
Value-function iteration \(V(s)\) on a discrete grid wherever the grid is dense ~4–6 states low-dim problems with occasionally-binding constraints
Projection (Judd) Chebyshev / splines on a tensor grid interior of the state domain; basis-dependent ~6–8 states medium-dim models with good structural properties
Parameterized expectations the conditional expectation as a polynomial where the polynomial fits the conditional ~6 states stochastic models where that expectation is the object
DEQN-JAX (this framework) a neural network, residuals on the simulated ergodic set wherever training samples reach no hard grid limit; exercised to ~13 states (disaster) higher-dim stochastic models with kinks, rare events, nonlinearities
PINN-HJB / KFE value function / density on continuous state via a PDE residual wherever collocation points reach ~4–6 continuous states continuous-time heterogeneous-agent (Aiyagari-class) models

The bottom three rows are all global. DEQN is the member that scales in the state dimension without a tensor grid and keeps the kinks — the four selling points below.

What the DEQN row buys you

  • Kinks stay kinked

    ZLB, borrowing limits, irreversible investment enter as Fischer–Burmeister complementarity residuals — solved globally, not linearized away at the steady state. Perturbation misses these entirely.

  • No tensor-grid curse

    The network plays the basis-function role of a projection method, but many state dimensions stay tractable — no grid to explode under the curse of dimensionality.

  • Composes with Dynare

    A first-order Blanchard–Kahn linearization — computed in-framework via QZ, or imported from Dynare — warm-starts and anchors the solve. You keep your perturbation workflow.

  • Accuracy you'd quote

    Reported as the distribution of relative Euler errors (errREE) on the ergodic set — the number you already put in a paper, not a black-box loss.

When DEQN earns its place vs Dynare specifically

Reach for DEQN when Dynare can't

  • Occasionally-binding constraints. ZLB, borrowing limits, irreversibility — first/second-order perturbation misses the kink. DEQN handles it natively, as Fischer–Burmeister residuals solved globally. The shipped examples that show this: bm_labor_constrained (labor cap), irbc (2-country irreversibility), olg_lifecycle (6-generation borrowing constraints).
  • Rare disasters / fat tails. A 1% disaster probability moves the ergodic distribution and the pricing kernel; a Taylor truncation underweights the tail. DEQN integrates the full shock distribution by Monte Carlo or Gauss–Hermite quadrature.
  • Higher state dimensions. Past ~10 states, perturbation loses accuracy far from the SS while VFI and projection collapse under the curse. DEQN is smooth interpolation regardless of \(d\).
  • Non-local counterfactuals. "What happens at 3σ from the steady state?" is exactly where linearization fails.

Reach for Dynare (or something else) when…

  • a first-order perturbation already answers your question — it is faster, proven, and the profession's common denominator;
  • you need Bayesian estimation — Dynare evaluates the likelihood and samples the posterior; DEQN-JAX solves a calibrated model, it does not estimate one;
  • you need a determinacy / equilibrium-selection guarantee or certified error bounds — DEQN provides neither (see the two limits below).

Two honest limits — stated up front, not in a footnote

  • A low residual is necessary but not sufficient. Like any nonlinear global solver, DEQN can settle on the wrong equilibrium branch, and nothing in the framework enforces equilibrium selection. There is no global analogue of the local Blanchard–Kahn saddle-path condition — this is a multiplicity / selection gap, not a "Blanchard–Kahn" criterion (BK is local and linear).
  • No analytic error bounds. Accuracy is measured (the errREE distribution), not proven by a theorem. Quote the number; don't assume it.

Composes with Dynare

DEQN-JAX extends your perturbation workflow — it does not ask you to throw it out. The first-order linearization is the warm start and the anchor; DEQN refines it into a global, nonlinear rule; you cross-check the result back against Dynare near the steady state.

flowchart LR
    DY["Perturb in Dynare<br/>(or in-framework QZ):<br/>first-order Blanchard&ndash;Kahn rule P"]
    DY -->|"warm_start_linearize /<br/>warm_start_dynare"| WS["Initialize: the policy<br/>IS the BK solution at step 0"]
    WS -->|"composite_loss.anchor_weight"| TR["Train against the<br/>nonlinear residuals,<br/>kinks intact"]
    TR --> XV["Cross-check: near-SS<br/>policy slope vs Dynare's P<br/>(Jacobian-match diagnostic)"]
    XV --> ACC["errREE on the ergodic path<br/>&mdash; the certificate you report"]
The recipe, step by step

A typical DEQN-in-a-paper recipe:

  1. Perturb in Dynare.
  2. Import the Blanchard–Kahn P matrix — warm_start_linearize to compute it in-framework via QZ, or warm_start_dynare to read a Dynare solution.
  3. Anchor the nonlinear solve to it (composite_loss.anchor_weight).
  4. Cross-validate near-SS behavior against Dynare (the Jacobian-match diagnostic).
  5. Report the errREE distribution on the ergodic path.

The LinearPlusMLP network bakes this in: the policy is the BK linear rule plus a zero-initialized correction, so at initialization the policy is the first-order solution and training can only improve on a correct local floor.

Where to go next

  • See worked models

    The constraint examples that show the sell — each with its measured errREE certificate.

    Gallery

  • Pick your method

    Networks, optimizers, expectations, diagnostics — and when (and when not) to reach for each.

    Method Zoo

  • Write your own model

    Declare states, equilibrium equations, transition, calibration — as data.

    Implementing a model

  • New to the method?

    A one-page orientation aimed at economists, before any code.

    What is DEQN?

Scope — what's in, what's out

In scope:

  • Discrete-time recursive general-equilibrium models with finite-dimensional state.
  • Any number of representative or finite-count agents (OLG with \(A\) generations; multi-country RBC — both shipped).
  • Shocks: continuous (Gaussian → lognormal / AR(1)) or discrete i.i.d., integrated by Monte Carlo or Gauss–Hermite quadrature.
  • Occasionally-binding constraints via Fischer–Burmeister complementarity residuals (bm_labor_constrained, irbc, olg_lifecycle).
  • Warm-starting / anchoring from a linearized solution for disaster-risk and kink settings.

Out of scope:

  • Continuous-time HJB + KFE models (Aiyagari / Krusell–Smith with a distributional state evolving under a Kolmogorov-forward PDE). These are the natural fit for PINN-HJB / finite-difference PDE solvers, not DEQN. A sibling PINN-HJB-KFE project in this research group complements DEQN — DEQN solves algebraic equilibrium conditions at sampled states; PINN-HJB solves PDEs on a discretized continuous state.
  • Mean-field games and any model whose state includes a measure evolving under a continuity equation.
  • Bayesian estimation. This framework solves a calibrated model; it does not evaluate a dataset's likelihood. Use Dynare or the estimation literature.
Against a hand-rolled implementation

DEQN-JAX is a JAX/Equinox reimplementation and extension of the Deep Equilibrium Nets method of Azinovic, Gaegauf & Scheidegger (2022). It assumes you already know the object you are training. What it adds over a single-model, hand-rolled implementation:

DEQN-JAX Hand-rolled JAX / PyTorch
Swap optimizer / network / expectation config change rewrite the loop
Batched model / variant comparison config-driven, built-in per-script
MC ↔ quadrature expectations config toggle hand-coded
Composite loss (anchor + Jacobian + barriers) config toggle reimplement
Diagnostic suite (errREE, IRFs, ergodic moments) shared across models per-script
Single JIT boundary yes depends on the author

The packaged models in src/deqn_jax/models/ are reference implementations meant to be read, forked, and extended — not a fixed catalogue. The full menu of swappable parts lives in the Method Zoo; the live registries are always the source of truth (uv run deqn-jax list, uv run deqn-jax optimizers).

Lineage & attribution

A JAX/Equinox reimplementation and extension of the Deep Equilibrium Nets methodology of Simon Scheidegger and collaborators; all credit for the original method belongs to the upstream authors.

  • Azinovic, M., Gaegauf, L., Scheidegger, S. (2022). Deep Equilibrium Nets. International Economic Review 63(4), 1471–1525.
  • Scheidegger, S., Bilionis, I. (2019). Machine learning for high-dimensional dynamic stochastic economies. Journal of Computational Science 33, 68–82.

This reimplementation migrates the approach to JAX + Equinox and adds architectural priors (LinearPlusMLP) and composite-loss terms. Full references on the home page.